HVT Scoring Cells with Layers using scoreLayeredHVT

Zubin Dowlaty, Srinivasan Sudarsanam, Somya Shambhawi

2024-02-12

1. Abstract

The HVT package is a collection of R functions to facilitate building topology preserving maps for rich multivariate data analysis. Tending towards a big data preponderance, a large number of rows. A collection of R functions for this typical workflow is organized below:

  1. Data Compression: Vector quantization (VQ), HVQ (hierarchical vector quantization) using means or medians. This step compresses the rows (long data frame) using a compression objective.

  2. Data Projection: Dimension projection of the compressed cells to 1D,2D or Interactive surface plot with the Sammons Non-linear Algorithm. This step creates topology preserving map (also called as embedding) coordinates into the desired output dimension.

  3. Tessellation: Create cells required for object visualization using the Voronoi Tessellation method, package includes heatmap plots for hierarchical Voronoi tessellations (HVT). This step enables data insights, visualization, and interaction with the topology preserving map. Useful for semi-supervised tasks.

  4. Scoring: Scoring new data sets and recording their assignment using the map objects from the above steps, in a sequence of maps if required.

2. Example : HVT with the Torus dataset

In this section, we will see how we can use the package to visualize multidimensional data by projecting them to two dimensions using Sammon’s projection and further used for Scoring

Data Understanding

First of all, let us see how to generate data for torus. We are using a library geozoo for this purpose. Geo Zoo (stands for Geometric Zoo) is a compilation of geometric objects ranging from three to 10 dimensions. Geo Zoo contains regular or well-known objects, eg cube and sphere, and some abstract objects, e.g. Boy’s surface, Torus and Hyper-Torus.

Here, we will generate a 3D torus (a torus is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle) with 12000 points.

Raw Torus Dataset

The torus dataset includes the following columns:

Lets, explore the raw torus dataset containing 12000 points. For the sake of brevity we are displaying first 6 rows.

set.seed(240)
# Here p represents dimension of object
# n represents number of points
torus <- geozoo::torus(p = 3,n = 12000)
torus_df <- data.frame(torus$points)
colnames(torus_df) <- c("x","y","z")
torus_df <- torus_df %>% round(4)
Table(head(torus_df))
x y z
-2.6282 0.5656 -0.7253
-1.4179 -0.8903 0.9455
-1.0308 1.1066 -0.8731
1.8847 0.1895 0.9944
-1.9506 -2.2507 0.2071
-1.4824 0.9229 0.9672

Now, let us check the structure of the data and analyse its summary.

str(torus_df)
#> 'data.frame':    12000 obs. of  3 variables:
#>  $ x: num  -2.63 -1.42 -1.03 1.88 -1.95 ...
#>  $ y: num  0.566 -0.89 1.107 0.19 -2.251 ...
#>  $ z: num  -0.725 0.946 -0.873 0.994 0.207 ...
summary(torus_df)
#>        x                   y                  z            
#>  Min.   :-2.997700   Min.   :-2.99930   Min.   :-1.000000  
#>  1st Qu.:-1.149025   1st Qu.:-1.11332   1st Qu.:-0.711950  
#>  Median :-0.007000   Median : 0.01305   Median : 0.015300  
#>  Mean   :-0.001444   Mean   : 0.01035   Mean   : 0.004423  
#>  3rd Qu.: 1.140325   3rd Qu.: 1.13373   3rd Qu.: 0.718550  
#>  Max.   : 2.999500   Max.   : 2.99930   Max.   : 1.000000

Let us first split the data into train and test. We will select the first 80% of the data for training and remaining as testing.

num_rows <- nrow(torus_df)
num_train_rows <- round(0.8 * num_rows)
num_test_rows <- num_rows - num_train_rows

trainTorus_data <- torus_df[1:num_train_rows, ]
testTorus_data <- torus_df[(num_train_rows + 1):(num_train_rows + num_test_rows), ]

Training Dataset

Now, lets have a look at the selected training dataset containing (9600 data points). For the sake of brevity we are displaying first six rows.

rownames(testTorus_data) <- NULL
Table(head(trainTorus_data))
x y z
-2.6282 0.5656 -0.7253
-1.4179 -0.8903 0.9455
-1.0308 1.1066 -0.8731
1.8847 0.1895 0.9944
-1.9506 -2.2507 0.2071
-1.4824 0.9229 0.9672

Now let’s have a look at structure and summary of the training data.

str(trainTorus_data)
#> 'data.frame':    9600 obs. of  3 variables:
#>  $ x: num  -2.63 -1.42 -1.03 1.88 -1.95 ...
#>  $ y: num  0.566 -0.89 1.107 0.19 -2.251 ...
#>  $ z: num  -0.725 0.946 -0.873 0.994 0.207 ...
summary(trainTorus_data)
#>        x                  y                  z            
#>  Min.   :-2.99770   Min.   :-2.99930   Min.   :-1.000000  
#>  1st Qu.:-1.16003   1st Qu.:-1.11512   1st Qu.:-0.712350  
#>  Median :-0.02605   Median : 0.01015   Median : 0.009950  
#>  Mean   :-0.01068   Mean   : 0.01097   Mean   : 0.004201  
#>  3rd Qu.: 1.13455   3rd Qu.: 1.13875   3rd Qu.: 0.718575  
#>  Max.   : 2.99950   Max.   : 2.99930   Max.   : 1.000000

Testing Dataset

Now, lets have a look at the randomly selected testing dataset containing (2400 data points). For the sake of brevity we are displaying first six rows.

rownames(testTorus_data) <- NULL
Table(head(testTorus_data))
x y z
-1.8031 1.5092 0.9362
1.1817 -1.0655 -0.9126
0.9942 -1.2500 -0.9153
1.0669 -0.0514 -0.3627
0.5570 -0.8837 -0.2954
0.8776 -0.4958 -0.1259

Now let’s have a look at structure and summary of the training data.

str(testTorus_data)
#> 'data.frame':    2400 obs. of  3 variables:
#>  $ x: num  -1.803 1.182 0.994 1.067 0.557 ...
#>  $ y: num  1.5092 -1.0655 -1.25 -0.0514 -0.8837 ...
#>  $ z: num  0.936 -0.913 -0.915 -0.363 -0.295 ...
summary(testTorus_data)
#>        x                  y                   z            
#>  Min.   :-2.99770   Min.   :-2.994800   Min.   :-1.000000  
#>  1st Qu.:-1.11670   1st Qu.:-1.105300   1st Qu.:-0.708175  
#>  Median : 0.08245   Median : 0.020150   Median : 0.037750  
#>  Mean   : 0.03551   Mean   : 0.007855   Mean   : 0.005307  
#>  3rd Qu.: 1.15878   3rd Qu.: 1.114550   3rd Qu.: 0.717050  
#>  Max.   : 2.99810   Max.   : 2.985900   Max.   : 1.000000

3. Map A : Base Compressed Map

Let us try to visualize the compressed Map A from the flow diagram below.

Figure 1: Data Segregation with highlighted bounding box in red around compressed map A

Figure 1: Data Segregation with highlighted bounding box in red around compressed map A

This package can perform vector quantization using the following algorithms -

For more information on vector quantization, refer the following link.

The trainHVT function constructs highly compressed hierarchical Voronoi tessellations. The raw data is first scaled and this scaled data is supplied as input to the vector quantization algorithm. The vector quantization algorithm compresses the dataset until a user-defined compression percentage/rate is achieved using a parameter called quantization error which acts as a threshold and determines the compression percentage. It means that for a given user-defined compression percentage we get the ‘n’ number of cells, then all of these cells formed will have a quantization error less than the threshold quantization error.

Let’s try to comprehend the trainHVT function first before moving ahead.

trainHVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  normalize = TRUE,
  diagnose = FALSE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters of trainHVT function have been explained below:

The output of trainHVT function (list of 6 elements) have been explained below:

We will use the trainHVT function to compress our data while preserving essential features of the dataset. Our goal is to achieve data compression upto atleast 80%. In situations where the compression ratio does not meet the desired target, we can explore adjusting the model parameters as a potential solution. This involves making modifications to parameters such as the quantization error threshold or increasing the number of cells and then rerunning the trainHVT function again.

As this is already done in HVT Vignette: please refer for more information.

Model Parameters

set.seed(240)
torus_mapA <- trainHVT(
  trainTorus_data,
  n_cells = 900,
  depth = 1,
  quant.err = 0.1,
  projection.scale = 10,
  normalize = FALSE,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s check the compression summary for torus.

compressionSummaryTable(torus_mapA[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 900 757 0.84 n_cells: 900 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

We successfully compressed 84% of the data using n_cells parameter as 900, the next step involves performing data projection on the compressed data. In this step, the compressed data will be transformed and projected onto a lower-dimensional space to visualize and analyze the data in a more manageable form.

As per the manual, torus_mapA[[3]] gives us detailed information about the hierarchical vector quantized data. torus_mapA[[3]][['summary']] gives a nice tabular data containing no of points, Quantization Error and the codebook.

The datatable displayed below is the summary from torus_mapA showing Cell.ID, Centroids and Quantization Error for each of the 900 cells.

summaryTable(torus_mapA[[3]]$summary,scroll = TRUE,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error x y z
1 1 1 15 874 0.1 2.94 0.48 -0.19
1 1 2 11 257 0.07 -1.05 1.18 0.90
1 1 3 10 497 0.05 0.14 -1.10 -0.46
1 1 4 8 540 0.07 0.30 -1.48 -0.87
1 1 5 13 251 0.12 -0.55 2.94 0.11
1 1 6 11 324 0.08 -0.83 -1.62 0.98
1 1 7 18 814 0.1 2.25 -0.47 0.95
1 1 8 6 38 0.06 -2.79 0.40 -0.57
1 1 9 8 661 0.04 1.28 0.29 -0.72
1 1 10 10 800 0.1 2.09 1.55 0.79
1 1 11 4 389 0.03 -0.50 0.87 -0.08
1 1 12 11 886 0.09 2.81 -0.68 0.44
1 1 13 12 202 0.09 -1.35 -2.09 -0.87
1 1 14 12 817 0.1 2.46 0.60 -0.84
1 1 15 9 900 0.09 2.49 -1.66 0.05
1 1 16 14 492 0.07 0.32 1.10 0.52
1 1 17 9 650 0.06 1.20 0.94 0.88
1 1 18 11 796 0.07 1.66 -1.68 0.93
1 1 19 11 130 0.1 -2.01 -0.91 0.98
1 1 20 14 856 0.13 2.71 0.07 0.69
1 1 21 15 729 0.12 1.22 2.44 -0.67
1 1 22 18 482 0.07 0.21 1.93 1.00
1 1 23 10 89 0.07 -2.38 -1.04 -0.80
1 1 24 12 726 0.08 1.48 1.62 -0.98
1 1 25 18 858 0.11 2.66 1.30 0.24
1 1 26 8 699 0.07 1.52 0.35 -0.89
1 1 27 14 291 0.07 -0.95 -2.01 -0.97
1 1 28 18 887 0.11 2.95 -0.29 -0.23
1 1 29 10 308 0.06 -1.00 0.23 0.23
1 1 30 8 195 0.06 -1.58 -0.04 0.91
1 1 31 16 377 0.08 -0.67 -1.74 -0.99
1 1 32 12 714 0.07 1.15 -1.56 -1.00
1 1 33 8 149 0.08 -1.74 -1.88 -0.82
1 1 34 11 478 0.05 0.19 0.99 0.12
1 1 35 9 518 0.06 0.51 1.12 0.64
1 1 36 8 898 0.09 2.45 -1.67 -0.24
1 1 37 14 32 0.11 -2.07 2.14 0.17
1 1 38 10 465 0.06 -0.08 -1.09 -0.41
1 1 39 14 476 0.06 -0.08 -1.55 -0.89
1 1 40 10 449 0.06 -0.05 1.00 -0.06
1 1 41 9 288 0.05 -1.12 -0.26 -0.53
1 1 42 14 116 0.13 -1.96 -1.95 -0.64
1 1 43 11 87 0.11 -2.44 -0.12 -0.89
1 1 44 9 437 0.05 -0.30 -1.10 -0.51
1 1 45 13 666 0.12 0.30 -2.97 0.03
1 1 46 7 414 0.09 -0.10 2.96 -0.27
1 1 47 14 595 0.08 0.86 -0.56 0.21
1 1 48 2 622 0.03 0.96 1.37 -0.95
1 1 49 8 132 0.07 -2.00 -0.63 0.99
1 1 50 15 347 0.09 -0.79 -1.01 0.69
1 1 51 9 432 0.07 -0.19 1.15 -0.55
1 1 52 6 270 0.07 -0.93 -2.55 -0.70
1 1 53 21 167 0.11 -1.53 1.57 -0.97
1 1 54 13 861 0.09 2.81 1.04 0.07
1 1 55 12 305 0.07 -1.05 -0.56 -0.58
1 1 56 10 479 0.05 0.17 1.07 -0.41
1 1 57 10 304 0.06 -1.01 0.33 0.34
1 1 58 14 421 0.09 -0.17 1.42 0.81
1 1 59 12 282 0.07 -1.09 0.79 -0.76
1 1 60 8 580 0.05 0.80 1.27 -0.86
1 1 61 11 576 0.1 0.24 -2.57 0.81
1 1 62 5 582 0.05 0.77 1.46 -0.94
1 1 63 9 266 0.05 -1.22 -0.15 -0.63
1 1 64 12 636 0.09 0.73 -1.49 -0.94
1 1 65 9 503 0.07 0.16 -1.41 0.81
1 1 66 19 259 0.08 -1.19 -0.48 0.69
1 1 67 23 560 0.08 0.63 -0.82 -0.26
1 1 68 11 642 0.05 1.17 0.45 -0.66
1 1 69 13 755 0.1 0.89 -2.64 0.61
1 1 70 12 570 0.08 0.86 0.99 0.72
1 1 71 10 283 0.06 -1.07 0.50 0.57
1 1 72 14 542 0.07 0.43 -1.08 -0.54
1 1 73 8 353 0.1 -0.39 2.76 -0.60
1 1 74 9 725 0.1 1.62 1.15 -0.99
1 1 75 7 416 0.06 -0.27 1.02 0.32
1 1 76 10 346 0.07 -0.79 0.63 -0.13
1 1 77 11 375 0.05 -0.60 0.90 -0.39
1 1 78 6 269 0.06 -0.97 -1.89 0.99
1 1 79 9 532 0.04 0.58 0.82 -0.06
1 1 80 16 237 0.09 -1.13 1.49 -0.99
1 1 81 13 372 0.07 -0.46 1.43 0.87
1 1 82 9 720 0.05 1.58 1.29 1.00
1 1 83 18 471 0.1 -0.23 -2.98 -0.04
1 1 84 14 875 0.1 2.93 0.37 0.28
1 1 85 13 888 0.1 2.85 -0.68 -0.34
1 1 86 15 271 0.06 -1.15 -0.17 0.55
1 1 87 7 512 0.04 0.44 0.91 -0.17
1 1 88 7 292 0.04 -1.10 -0.12 -0.44
1 1 89 7 92 0.06 -2.34 0.76 -0.88
1 1 90 8 752 0.07 1.41 2.30 -0.71
1 1 91 11 648 0.08 1.24 0.36 0.70
1 1 92 7 548 0.06 0.70 0.73 0.15
1 1 93 13 741 0.11 1.23 2.69 0.27
1 1 94 19 811 0.1 2.03 -1.22 0.93
1 1 95 13 746 0.08 0.94 -2.45 -0.77
1 1 96 10 880 0.12 2.66 -1.09 0.48
1 1 97 7 262 0.07 -0.54 2.92 -0.23
1 1 98 10 713 0.08 1.41 -0.85 -0.93
1 1 99 11 768 0.07 1.52 -1.59 -0.98
1 1 100 11 420 0.08 -0.36 -1.35 0.80
1 1 101 16 829 0.12 2.51 1.10 0.66
1 1 102 11 847 0.1 1.74 -2.27 -0.50
1 1 103 9 2 0.08 -2.99 0.21 -0.05
1 1 104 8 43 0.07 -2.53 1.31 -0.52
1 1 105 13 222 0.09 -0.85 2.54 -0.73
1 1 106 5 264 0.04 -1.15 0.32 0.60
1 1 107 8 686 0.06 1.08 -1.38 0.97
1 1 108 9 13 0.08 -2.88 0.45 0.41
1 1 109 11 659 0.06 1.27 0.01 0.69
1 1 110 15 184 0.06 -1.53 1.13 0.99
1 1 111 9 246 0.06 -1.33 0.17 -0.75
1 1 112 6 552 0.04 0.71 0.76 0.29
1 1 113 10 210 0.1 -1.12 -2.44 0.72
1 1 114 15 547 0.09 0.53 2.14 0.97
1 1 115 13 435 0.07 -0.29 -1.91 1.00
1 1 116 8 95 0.07 -2.38 -0.84 -0.85
1 1 117 6 46 0.04 -2.76 -0.64 -0.55
1 1 118 13 86 0.08 -2.35 -0.92 0.85
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1 1 120 13 643 0.06 0.99 1.60 -0.99
1 1 121 15 594 0.09 0.74 -1.04 0.69
1 1 122 13 80 0.09 -2.15 1.55 0.75
1 1 123 6 404 0.08 -0.17 2.75 -0.65
1 1 124 8 586 0.06 0.90 0.87 -0.66
1 1 125 12 19 0.1 -2.78 1.03 -0.22
1 1 126 7 111 0.08 -1.75 2.05 -0.71
1 1 127 6 24 0.07 -2.18 2.05 -0.09
1 1 128 12 680 0.09 0.77 -2.00 0.98
1 1 129 8 307 0.05 -0.96 0.63 0.53
1 1 130 12 407 0.08 -0.50 -2.07 -0.99
1 1 131 9 345 0.06 -0.79 0.73 -0.36
1 1 132 9 53 0.06 -2.63 0.13 0.77
1 1 133 10 84 0.11 -2.40 0.43 0.89
1 1 134 9 879 0.08 2.15 -1.95 0.43
1 1 135 10 620 0.06 1.07 -0.07 0.37
1 1 136 12 805 0.09 2.30 0.23 -0.95
1 1 137 12 588 0.07 0.32 -2.19 -0.97
1 1 138 13 148 0.1 -1.93 0.54 -1.00
1 1 139 20 362 0.08 -0.74 -0.71 0.23
1 1 140 7 775 0.08 1.16 -2.45 0.70
1 1 141 8 272 0.06 -1.09 1.14 -0.90
1 1 142 8 771 0.09 1.71 1.94 0.80
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1 1 147 12 607 0.08 1.04 0.63 0.62
1 1 148 7 494 0.08 0.07 -1.43 -0.82
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1 1 151 8 11 0.07 -2.82 1.01 0.01
1 1 152 7 514 0.05 0.47 0.92 0.25
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1 1 226 15 700 0.1 1.01 2.64 0.55
1 1 227 14 779 0.11 1.87 -0.82 1.00
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1 1 232 6 794 0.06 1.76 2.13 0.64
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1 1 234 8 778 0.06 1.97 -0.36 1.00
1 1 235 12 827 0.11 2.26 -0.95 0.88
1 1 236 14 554 0.06 0.56 -0.83 0.05
1 1 237 8 706 0.06 1.55 0.75 -0.96
1 1 238 9 107 0.1 -2.31 0.16 -0.94
1 1 239 14 539 0.08 0.36 -1.45 0.86
1 1 240 6 15 0.06 -2.92 0.23 -0.37
1 1 241 7 374 0.05 -0.66 -0.91 0.49
1 1 242 9 263 0.08 -1.23 0.26 -0.67
1 1 243 7 206 0.05 -1.51 0.79 -0.95
1 1 244 10 415 0.04 -0.29 0.97 0.14
1 1 245 19 609 0.11 0.42 -2.16 0.97
1 1 246 7 864 0.07 1.82 -2.32 -0.30
1 1 247 8 275 0.08 -1.18 -0.68 -0.77
1 1 248 13 390 0.09 -0.52 -1.76 0.98
1 1 249 11 373 0.08 -0.35 2.38 -0.91
1 1 250 11 26 0.08 -2.42 1.63 -0.39
1 1 251 5 550 0.04 0.70 0.72 -0.04
1 1 252 8 543 0.05 0.54 1.86 1.00
1 1 253 10 673 0.1 0.48 -2.58 -0.77
1 1 254 13 583 0.06 0.74 -0.84 -0.47
1 1 255 14 601 0.09 0.92 -0.39 -0.02
1 1 256 7 30 0.07 -2.69 -1.15 0.37
1 1 257 10 707 0.08 0.98 -1.84 -0.99
1 1 258 12 753 0.1 1.85 0.42 0.99
1 1 259 8 742 0.08 1.22 2.74 -0.02
1 1 260 11 287 0.1 -0.57 2.63 -0.71
1 1 261 13 200 0.08 -1.48 -1.09 0.98
1 1 262 10 820 0.12 1.52 -2.34 0.61
1 1 263 14 281 0.12 -0.55 2.47 0.84
1 1 264 8 102 0.09 -1.97 1.64 -0.82
1 1 265 7 667 0.05 0.62 -2.13 -0.97
1 1 266 7 681 0.06 1.39 0.88 -0.93
1 1 267 10 365 0.08 -0.43 2.13 -0.98
1 1 268 13 447 0.11 -0.31 -2.62 -0.76
1 1 269 11 48 0.09 -2.68 0.42 0.70
1 1 270 10 124 0.08 -2.14 -1.31 -0.85
1 1 271 13 533 0.06 0.59 0.95 0.46
1 1 272 13 687 0.13 0.96 2.68 -0.52
1 1 273 16 79 0.13 -2.24 -1.79 -0.48
1 1 274 13 42 0.1 -2.53 1.18 0.59
1 1 275 12 327 0.07 -0.90 0.57 -0.35
1 1 276 10 892 0.09 2.81 -0.94 0.26
1 1 277 16 252 0.08 -1.18 -1.03 0.90
1 1 278 14 417 0.1 -0.40 -2.47 0.86
1 1 279 6 289 0.05 -1.10 0.05 -0.44
1 1 280 8 777 0.07 2.02 1.26 0.92
1 1 281 7 885 0.11 2.99 0.05 -0.08
1 1 282 8 230 0.05 -1.41 -0.86 -0.94
1 1 283 10 36 0.09 -2.21 1.91 -0.37
1 1 284 15 311 0.05 -1.00 -0.05 0.06
1 1 285 9 81 0.08 -2.34 1.23 -0.76
1 1 286 15 838 0.11 2.61 0.22 -0.78
1 1 287 7 189 0.04 -1.57 0.65 0.96
1 1 288 13 844 0.12 2.45 -0.68 0.83
1 1 289 10 495 0.07 0.31 1.51 0.89
1 1 290 13 549 0.06 0.69 0.75 -0.19
1 1 291 9 73 0.08 -2.03 1.95 -0.58
1 1 292 10 198 0.06 -1.63 -0.36 -0.94
1 1 293 13 691 0.06 1.46 0.00 0.84
1 1 294 8 754 0.08 1.23 -1.92 0.96
1 1 295 9 534 0.04 0.59 0.84 0.20
1 1 296 12 323 0.08 -0.85 0.95 0.69
1 1 297 16 522 0.12 0.39 2.68 0.70
1 1 298 5 662 0.03 1.28 0.52 -0.78
1 1 299 18 553 0.08 0.47 -1.20 0.71
1 1 300 18 867 0.13 2.22 -1.65 0.64
1 1 301 8 430 0.1 -0.39 -2.33 -0.93
1 1 302 7 490 0.05 0.30 0.95 -0.03
1 1 303 5 678 0.06 1.35 0.11 -0.77
1 1 304 10 120 0.12 -1.80 1.82 -0.83
1 1 305 13 584 0.07 0.77 -0.84 0.50
1 1 306 12 719 0.13 0.58 -2.86 -0.37
1 1 307 7 51 0.08 -2.75 -0.11 -0.65
1 1 308 17 356 0.07 -0.79 -0.63 -0.11
1 1 309 9 558 0.1 0.48 2.95 -0.04
1 1 310 9 555 0.06 0.30 -1.71 -0.96
1 1 311 13 173 0.07 -1.77 0.18 -0.97
1 1 312 9 54 0.12 -1.89 2.31 -0.12
1 1 313 14 709 0.08 1.42 -0.76 0.92
1 1 314 11 708 0.08 1.55 0.09 -0.89
1 1 315 7 227 0.07 -1.39 0.22 0.80
1 1 316 9 883 0.1 2.10 -2.11 0.22
1 1 317 8 401 0.1 -0.16 2.99 0.09
1 1 318 13 96 0.08 -1.56 2.43 0.45
1 1 319 6 364 0.04 -0.68 0.74 -0.02
1 1 320 8 521 0.06 0.13 -1.79 -0.98
1 1 321 2 546 0.01 0.64 1.04 -0.63
1 1 322 16 459 0.08 -0.12 -1.47 0.85
1 1 323 10 510 0.07 -0.01 -2.28 -0.96
1 1 324 10 387 0.08 -0.32 1.71 0.96
1 1 325 4 383 0.03 -0.54 0.84 -0.04
1 1 326 14 363 0.11 -0.36 2.23 0.96
1 1 327 17 297 0.08 -1.04 -0.37 0.43
1 1 328 7 869 0.11 2.75 -0.49 -0.60
1 1 329 10 98 0.1 -2.25 -1.09 0.86
1 1 330 14 544 0.07 0.49 -0.94 -0.35
1 1 331 14 242 0.07 -1.29 0.91 -0.90
1 1 332 14 590 0.06 0.96 0.51 0.41
1 1 333 13 354 0.12 -0.34 2.61 0.77
1 1 334 9 12 0.07 -2.89 0.76 -0.09
1 1 335 9 840 0.09 2.52 -0.30 -0.84
1 1 336 10 545 0.06 0.58 1.31 -0.82
1 1 337 9 872 0.12 2.74 -0.41 0.63
1 1 338 10 496 0.06 0.34 0.95 0.12
1 1 339 7 162 0.08 -1.33 -2.67 0.12
1 1 340 7 783 0.09 2.11 0.68 -0.97
1 1 341 6 228 0.05 -1.45 0.27 -0.85
1 1 342 11 409 0.06 -0.47 -1.18 -0.68
1 1 343 6 267 0.07 -1.22 -0.32 -0.67
1 1 344 6 767 0.06 1.97 0.41 -1.00
1 1 345 15 133 0.11 -1.49 2.24 -0.72
1 1 346 8 39 0.07 -2.78 -0.33 -0.59
1 1 347 10 443 0.05 -0.09 1.01 0.14
1 1 348 9 250 0.09 -0.63 2.81 -0.47
1 1 349 6 68 0.09 -1.89 2.13 0.51
1 1 350 7 839 0.11 1.51 -2.55 -0.24
1 1 351 11 388 0.06 -0.44 1.36 -0.82
1 1 352 10 694 0.08 1.24 1.86 0.97
1 1 353 11 76 0.09 -2.06 -2.14 0.25
1 1 354 10 795 0.1 1.15 -2.64 0.48
1 1 355 16 434 0.1 -0.01 2.56 0.82
1 1 356 13 10 0.07 -2.69 1.30 0.14
1 1 357 10 464 0.05 0.10 1.10 0.44
1 1 358 12 247 0.06 -1.33 0.00 -0.74
1 1 359 6 889 0.08 2.63 -1.31 -0.34
1 1 360 7 612 0.06 1.00 -0.21 -0.18
1 1 361 7 125 0.06 -2.07 0.37 0.99
1 1 362 8 878 0.09 1.96 -2.27 0.05
1 1 363 16 338 0.11 -0.50 1.96 1.00
1 1 364 12 652 0.06 1.01 -0.86 -0.74
1 1 365 15 793 0.09 2.16 0.54 0.97
1 1 366 5 61 0.06 -2.64 0.29 -0.76
1 1 367 15 891 0.12 2.94 -0.48 0.19
1 1 368 6 592 0.05 0.39 -1.96 -1.00
1 1 369 11 408 0.08 -0.43 -2.12 0.98
1 1 370 11 318 0.08 -0.97 -1.22 -0.90
1 1 371 14 74 0.07 -2.51 -0.28 0.85
1 1 372 9 734 0.07 0.90 -2.43 0.80
1 1 373 9 310 0.06 -1.00 0.12 0.11
1 1 374 13 35 0.11 -2.50 -1.63 -0.13
1 1 375 14 615 0.07 1.03 0.06 -0.27
1 1 376 18 527 0.07 0.20 -1.81 0.98
1 1 377 12 410 0.06 -0.45 -0.96 -0.35
1 1 378 9 398 0.06 -0.43 0.94 0.24
1 1 379 7 781 0.06 1.75 -1.23 0.99
1 1 380 9 688 0.07 0.67 -2.34 -0.90
1 1 381 13 135 0.12 -1.37 2.33 0.71
1 1 382 9 321 0.06 -0.96 0.41 -0.30
1 1 383 11 424 0.08 -0.39 -1.42 -0.85
1 1 384 8 397 0.06 -0.53 -0.86 0.14
1 1 385 11 169 0.09 -0.98 2.81 -0.21
1 1 386 11 868 0.08 2.82 -0.11 -0.56
1 1 387 9 854 0.09 2.40 -1.25 -0.71
1 1 388 9 809 0.09 1.31 -2.57 -0.45
1 1 389 6 7 0.05 -2.34 1.87 0.03
1 1 390 8 165 0.05 -1.76 -1.36 -0.97
1 1 391 6 682 0.07 1.41 0.63 -0.89
1 1 392 10 761 0.06 1.80 -0.39 -0.98
1 1 393 10 193 0.09 -1.19 -2.65 0.41
1 1 394 14 799 0.13 1.90 1.94 -0.69
1 1 395 10 704 0.06 1.38 1.56 1.00
1 1 396 5 295 0.07 -0.91 -2.30 -0.88
1 1 397 15 395 0.09 -0.38 1.22 0.69
1 1 398 11 216 0.07 -1.44 -0.18 0.83
1 1 399 12 221 0.05 -1.19 1.43 0.99
1 1 400 12 523 0.1 0.38 2.72 -0.66
1 1 401 11 215 0.06 -1.49 0.51 -0.90
1 1 402 5 569 0.07 0.57 2.82 -0.47
1 1 403 9 104 0.06 -2.30 0.55 -0.93
1 1 404 8 824 0.1 2.01 -1.52 0.85
1 1 405 11 214 0.09 -0.85 2.40 0.83
1 1 406 18 219 0.11 -0.95 2.32 -0.86
1 1 407 7 4 0.08 -2.96 -0.25 0.20
1 1 408 8 760 0.07 1.74 -0.65 -0.99
1 1 409 12 641 0.06 1.16 0.07 -0.54
1 1 410 7 261 0.06 -1.22 0.47 -0.72
1 1 411 7 371 0.06 -0.63 0.79 -0.15
1 1 412 15 613 0.07 0.98 -0.40 0.33
1 1 413 15 187 0.08 -1.69 -0.90 -0.99
1 1 414 15 677 0.08 1.33 -0.29 0.77
1 1 415 11 166 0.09 -1.71 -0.89 0.99
1 1 416 6 379 0.06 -0.59 0.81 0.10
1 1 417 16 212 0.06 -1.53 0.07 -0.88
1 1 418 15 224 0.11 -0.98 -2.82 0.11
1 1 419 12 359 0.07 -0.79 -0.81 -0.50
1 1 420 11 376 0.07 -0.68 -0.79 -0.30
1 1 421 9 143 0.09 -1.88 -1.24 0.96
1 1 422 15 656 0.08 0.77 -1.70 0.99
1 1 423 6 302 0.04 -1.03 0.44 -0.48
1 1 424 9 163 0.06 -1.82 -1.18 -0.98
1 1 425 11 431 0.07 -0.08 1.77 0.97
1 1 426 14 234 0.06 -1.34 -0.30 0.78
1 1 427 14 602 0.12 0.20 -2.81 -0.56
1 1 428 14 664 0.1 1.33 0.63 0.85
1 1 429 14 147 0.1 -1.62 -2.05 0.78
1 1 430 13 849 0.12 2.50 1.55 -0.32
1 1 431 10 561 0.05 0.81 0.59 0.04
1 1 432 8 183 0.07 -1.65 0.23 0.94
1 1 433 13 100 0.06 -2.24 0.57 0.95
1 1 434 14 573 0.07 0.75 -0.66 -0.04
1 1 435 11 749 0.09 1.80 0.05 -0.98
1 1 436 13 350 0.08 -0.82 -0.58 0.10
1 1 437 10 655 0.08 1.17 -0.31 -0.61
1 1 438 9 440 0.1 0.01 2.78 0.61
1 1 439 7 3 0.08 -2.99 -0.07 -0.14
1 1 440 13 231 0.07 -1.38 -1.09 -0.97
1 1 441 9 313 0.07 -1.01 -0.78 -0.69
1 1 442 12 848 0.11 2.30 1.91 0.05
1 1 443 15 220 0.06 -1.48 -0.57 -0.91
1 1 444 9 366 0.09 -0.62 -2.93 -0.01
1 1 445 8 22 0.09 -2.28 1.85 0.34
1 1 446 15 764 0.12 1.76 1.63 -0.91
1 1 447 14 6 0.1 -2.54 1.54 0.22
1 1 448 11 765 0.1 0.81 -2.88 0.05
1 1 449 19 604 0.06 0.99 0.36 -0.32
1 1 450 10 774 0.09 2.03 0.14 -1.00
1 1 451 8 788 0.08 1.53 2.57 -0.08
1 1 452 9 710 0.07 1.54 -0.25 0.90
1 1 453 11 157 0.09 -1.41 -2.49 0.49
1 1 454 8 816 0.08 1.79 -1.80 -0.84
1 1 455 13 850 0.08 1.88 -2.08 0.59
1 1 456 9 126 0.08 -1.34 2.65 -0.21
1 1 457 11 182 0.06 -1.72 0.44 -0.97
1 1 458 7 97 0.07 -2.29 0.00 0.95
1 1 459 6 188 0.05 -1.41 1.36 1.00
1 1 460 11 716 0.14 1.19 2.34 0.77
1 1 461 13 334 0.07 -0.87 -0.68 0.44
1 1 462 14 412 0.07 -0.40 -0.98 0.35
1 1 463 18 787 0.11 1.02 -2.77 -0.29
1 1 464 14 461 0.07 0.10 1.33 0.74
1 1 465 17 392 0.08 -0.44 1.03 0.48
1 1 466 12 803 0.08 2.02 1.84 0.68
1 1 467 10 631 0.07 1.13 0.79 0.78
1 1 468 14 498 0.06 0.18 -0.99 -0.08
1 1 469 11 692 0.06 1.35 -0.59 -0.85
1 1 470 11 320 0.07 -0.95 0.68 -0.55
1 1 471 15 587 0.08 0.64 -1.21 -0.77
1 1 472 13 357 0.08 -0.60 1.49 -0.92
1 1 473 16 833 0.1 1.86 -1.85 0.78
1 1 474 5 472 0.05 0.17 1.46 0.85
1 1 475 12 600 0.07 0.86 -0.69 0.44
1 1 476 7 91 0.1 -1.97 -2.05 0.53
1 1 477 13 303 0.06 -0.78 1.67 -0.98
1 1 478 11 823 0.1 1.36 -2.66 -0.04
1 1 479 9 515 0.06 0.30 -1.01 -0.31
1 1 480 12 8 0.09 -2.46 1.70 -0.10
1 1 481 11 563 0.07 0.84 0.86 0.60
1 1 482 14 294 0.05 -1.06 0.12 0.35
1 1 483 7 455 0.07 -0.19 -1.39 -0.80
1 1 484 10 393 0.04 -0.57 -0.82 0.00
1 1 485 9 832 0.07 2.37 -0.70 -0.88
1 1 486 14 504 0.08 0.20 -1.21 0.63
1 1 487 13 632 0.07 0.93 -0.95 0.74
1 1 488 16 853 0.14 2.78 0.93 -0.33
1 1 489 12 723 0.1 1.34 2.00 -0.91
1 1 490 12 526 0.09 0.39 2.41 -0.89
1 1 491 10 88 0.07 -2.28 -1.51 -0.67
1 1 492 13 58 0.11 -2.07 1.89 0.59
1 1 493 12 245 0.11 -0.92 -2.79 -0.32
1 1 494 7 64 0.05 -2.18 -2.06 0.10
1 1 495 10 702 0.08 1.44 1.29 -0.99
1 1 496 9 473 0.07 0.11 1.18 -0.58
1 1 497 8 205 0.08 -1.23 -2.02 0.93
1 1 498 11 568 0.06 0.63 -1.00 -0.58
1 1 499 10 329 0.06 -0.77 -2.51 -0.78
1 1 500 13 105 0.1 -1.56 2.45 -0.40

Now let us understand what each column in the above table means:

All the columns after this will contain centroids for each cell. They can also be called a codebook, which represents a collection of all centroids or codewords.

Now let’s try to understand plotHVT function. The parameters have been explained in detail below:

plotHVT <-(hvt.results, line.width, color.vec, pch1 = 21, palette.color = 6, centroid.size = 1.5, title = NULL, maxDepth = NULL, dataset, child.level, hmap.cols, previous_level_heatmap = TRUE, show.points = FALSE, asp = 1, ask = TRUE, tess.label = NULL, quant.error.hmap = NULL, n_cells.hmap = NULL, label.size = 0.5, sepration_width = 7, layer_opacity = c(0.5, 0.75, 0.99), dim_size = 1000, heatmap = '2Dhvt') 

Let’s plot the Voronoi tessellation for layer 1 (map A).

plotHVT(torus_mapA,
        line.width = c(0.4), 
        color.vec = c("#141B41"),
        centroid.size = 0.01,
        maxDepth = 1, heatmap = '2Dhvt') 
Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 900 cells in the dataset ’torus’

Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 900 cells in the dataset ’torus’

Heat Maps

We will now overlay all the features as heatmap over the Voronoi Tessellation plot for better visualization and identification of patterns, trends, and variations in the data.

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data for better visualization and interpretation of data patterns and distributions.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the torus data, allowing us to observe patterns and trends in the distribution of each of the features (x,y,z). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the torus data.


  plotHVT(
  torus_mapA,
  trainTorus_data,
  child.level = 1,
  hmap.cols = "x",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 900,
  heatmap = '2Dheatmap'
) 
Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’x’ in the ’torus’ dataset

Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’x’ in the ’torus’ dataset


  plotHVT(
  torus_mapA,
  trainTorus_data,
  child.level = 1,
  hmap.cols = "y",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 900,
  heatmap = '2Dheatmap'
) 
Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’y’ in the ’torus’ dataset

Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’y’ in the ’torus’ dataset


  plotHVT(
  torus_mapA,
  trainTorus_data,
  child.level = 1,
  hmap.cols = "z",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
  heatmap = '2Dheatmap'
) 
Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’z’ in the ’torus’ dataset

Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’z’ in the ’torus’ dataset

4. Map B : Compressed Novelty Map

Let us try to visualize the Map B from the flow diagram below.

Figure 10: Data Segregation with highlighted bounding box in red around map B

Figure 10: Data Segregation with highlighted bounding box in red around map B

In this section, we will manually figure out the novelty cells from the plotted torus_mapA and store it in identified_Novelty_cells variable.

Note: For manual selecting the novelty cells from map A, one can enhance its interactivity by adding plotly elements to the code. This will transform map A into an interactive plot, allowing users to actively engage with the data. By hovering over the centroids of the cells, a tag containing segment child information will be displayed. Users can explore the map by hovering over different cells and selectively choose the novelty cells they wish to consider. Added an image for reference.

Figure 11: Manually selecting novelty cells

Figure 11: Manually selecting novelty cells

The removeNovelty function removes the identified novelty cell(s) from the training dataset (containing 9600 datapoints) and stores those records separately.

It takes input as the cell number (Segment.Child) of the manually identified novelty cell(s) and the compressed HVT map (torus_mapA) with 900 cells. It returns a list of two items: data with novelty, and a subset of the data without novelty.

NOTE: As we are using torus data here, the identified novelty cells given are for demo purpose.

identified_Novelty_cells <<- c(478,448,626,220,374,644,442,197)   #as a example
output_list <- removeNovelty(identified_Novelty_cells, torus_mapA)
data_with_novelty <- output_list[[1]]
data_without_novelty <- output_list[[2]]

Let’s have a look at the data with novelty(containing 78 records). For the sake of brevity, we will only show the first 20 rows.

novelty_data <- data_with_novelty
novelty_data$Row.No <- row.names(novelty_data)
novelty_data <- novelty_data %>% dplyr::select("Row.No","Cell.ID","Cell.Number","x","y","z")
colnames(novelty_data) <- c("Row.No","Cell.ID","Segment.Child","x","y","z")
novelty_data %>% head(100) %>% 
  as.data.frame() %>%
  Table(scroll = TRUE, limit = 20)
Row.No Cell.ID Segment.Child x y z
1 857 197 2.4975 1.6226 0.2071
2 857 197 2.4936 1.6507 0.1380
3 857 197 2.5783 1.5276 0.0794
4 857 197 2.4381 1.7455 0.0552
5 857 197 2.4653 1.7095 0.0057
6 857 197 2.4886 1.5752 0.3264
7 857 197 2.5611 1.5621 -0.0136
8 857 197 2.5374 1.5475 0.2346
9 52 220 -2.2472 -1.9850 0.0571
10 52 220 -2.2326 -2.0003 -0.0687
11 52 220 -2.2476 -1.9798 -0.0977
12 52 220 -2.2374 -1.9752 -0.1755
13 52 220 -2.3188 -1.8912 0.1244
14 52 220 -2.3080 -1.9028 0.1321
15 52 220 -2.2446 -1.9892 0.0410
16 52 220 -2.4092 -1.7780 0.1069
17 52 220 -2.2843 -1.9121 0.2041
18 52 220 -2.3390 -1.8785 0.0041
19 35 374 -2.4310 -1.7577 -0.0142
20 35 374 -2.4425 -1.7417 -0.0149

4.1 Voronoi Tessellation with highlighted novelty cell

The plotNovelCells function is used to plot the Voronoi tessellation using the compressed HVT map (torus_mapA) containing 900 cells and highlights the identified novelty cell(s) i.e 8 cells (containing 78 records) in red on the map.

plotNovelCells(identified_Novelty_cells, torus_mapA,line.width = c(0.4),centroid.size = 0.01)
Figure 12: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

Figure 12: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

We pass the dataframe with novelty records (78 records) to trainHVT function along with other model parameters mentioned below to generate map B (layer2)

Model Parameters

colnames(data_with_novelty) <- c("Cell.ID","Segment.Child","x","y","z")
data_with_novelty <- data_with_novelty[,-1:-2]
torus_mapB <- list()
mapA_scale_summary = torus_mapA[[3]]$scale_summary
torus_mapB <- trainHVT(data_with_novelty,
                  n_cells = 11,   
                  depth = 1,
                  quant.err = 0.1,
                  projection.scale = 10,
                  normalize = FALSE,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans"
                  )

The datatable displayed below is the summary from map B (layer 2) showing Cell.ID, Centroids and Quantization Error for each of the 11 cells.

summaryTable(torus_mapB[[3]]$summary,scroll = TRUE,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error x y z
1 1 1 5 6 0.05 -2.01 -2.22 -0.05
1 1 2 5 3 0.08 1.75 -2.43 -0.08
1 1 3 6 10 0.06 -2.44 -1.73 -0.10
1 1 4 7 11 0.08 -2.55 -1.54 -0.16
1 1 5 9 1 0.09 2.28 1.94 0.02
1 1 6 11 4 0.1 1.36 -2.66 -0.04
1 1 7 3 9 0.04 -2.10 -2.13 -0.12
1 1 8 5 7 0.06 -2.33 -1.87 0.11
1 1 9 5 8 0.05 -2.24 -1.99 -0.05
1 1 10 11 5 0.1 0.81 -2.88 0.05
1 1 11 11 2 0.13 2.47 1.67 0.13

Now let’s check the compression summary for HVT (torus_mapB). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapB_compression_summary <- torus_mapB[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapB_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 11 9 0.82 n_cells: 11 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 82% of the cells have hit the quantization threshold error.Since we are successfully able to attain the desired compression percentage, so we will not further subdivide the cells

5. Map C : Compressed Map without Novelty

Let us try to visualize the compressed Map C from the flow diagram below.

Figure 13:Data Segregation with highlighted bounding box in red around compressed map C

Figure 13:Data Segregation with highlighted bounding box in red around compressed map C

5.1 Iteration 1:

With the Novelties removed, we construct another hierarchical Voronoi tessellation map C layer 2 on the data without Novelty (containing 9522 records) and below mentioned model parameters.

Model Parameters

torus_mapC <- list()
mapA_scale_summary = torus_mapA[[3]]$scale_summary
torus_mapC <- trainHVT(data_without_novelty,
                  n_cells = 10,
                  depth = 2,
                  quant.err = 0.1,
                  projection.scale = 10,
                  normalize = FALSE,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = FALSE,
                  scale_summary = mapA_scale_summary)

Now let’s check the compression summary for HVT (torus_mapC) where n_cell was set to 10. The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- torus_mapC[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 10 0 0 n_cells: 10 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 100 0 0 n_cells: 10 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 0% of the cells have hit the quantization threshold error in level 2

5.2 Iteration 2:

Since, we are yet to achive atleast 80% compression at depth 2. Let’s try to compress again using the below mentioned set of model parameters and the data without novelty (containing 9522 records).

Model Parameters

torus_mapC <- list()
torus_mapC <- trainHVT(data_without_novelty,
                  n_cells = 30,    
                  depth = 2,
                  quant.err = 0.1,
                  projection.scale = 10,
                  normalize = FALSE,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = FALSE,
                  scale_summary = mapA_scale_summary)

The datatable displayed below is the summary from map C (layer2). showing Cell.ID, Centroids and Quantization Error for each of the 924 cells.

summaryTable(torus_mapC[[3]]$summary,scroll = TRUE,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error x y z
1 1 1 273 224 0.54 -1.43 1.97 0.75
1 1 2 392 508 0.48 1.33 -0.76 -0.81
1 1 3 353 397 0.45 1.03 0.21 0.09
1 1 4 358 342 0.5 -0.84 1.19 -0.76
1 1 5 300 802 0.54 -1.59 -1.47 -0.87
1 1 6 395 287 0.56 0.55 1.02 -0.47
1 1 7 344 710 0.43 -0.45 -1.52 0.81
1 1 8 291 728 0.65 -1.98 -0.82 0.87
1 1 9 371 559 0.48 -1.49 -0.03 -0.80
1 1 10 289 160 0.53 2.40 0.81 0.65
1 1 11 367 556 0.51 0.60 -0.93 0.36
1 1 12 287 391 0.55 -2.36 1.35 -0.44
1 1 13 343 175 0.61 2.00 0.97 -0.83
1 1 14 275 368 0.54 2.72 -0.33 -0.39
1 1 15 292 81 0.61 0.58 2.32 -0.77
1 1 16 252 123 0.55 -0.94 2.48 -0.58
1 1 17 283 491 0.56 -2.13 0.66 0.84
1 1 18 268 100 0.58 -0.03 2.42 0.74
1 1 19 264 776 0.57 0.84 -2.23 -0.78
1 1 20 274 766 0.54 -2.73 -0.50 -0.34
1 1 21 299 763 0.58 0.79 -2.25 0.78
1 1 22 249 29 0.57 1.59 2.29 0.31
1 1 23 328 228 0.43 0.90 1.22 0.83
1 1 24 205 873 0.58 -1.65 -2.12 0.52
1 1 25 444 655 0.53 -0.26 -1.17 -0.52
1 1 26 270 676 0.55 2.20 -1.75 0.10
1 1 27 262 881 0.53 -0.57 -2.70 -0.28
1 1 28 483 536 0.47 -1.03 -0.14 0.30
1 1 29 314 459 0.52 1.79 -0.57 0.90
1 1 30 397 350 0.48 -0.51 1.04 0.50
2 1 1 3 195 0.04 -0.72 1.95 1.00
2 1 2 6 366 0.04 -1.56 1.28 1.00
2 1 3 13 182 0.18 -2.02 2.19 0.14
2 1 4 12 249 0.12 -1.73 1.85 0.84
2 1 5 13 236 0.13 -2.10 1.92 0.52
2 1 6 4 339 0.05 -1.71 1.43 0.97
2 1 7 11 296 0.06 -1.85 1.61 0.89
2 1 8 5 242 0.06 -1.12 1.76 0.99
2 1 9 12 191 0.12 -1.68 2.15 0.68
2 1 10 16 284 0.08 -1.54 1.60 0.97
2 1 11 17 231 0.09 -1.42 1.92 0.92
2 1 12 11 286 0.08 -2.07 1.65 0.76
2 1 13 5 129 0.07 -1.10 2.47 0.71
2 1 14 13 137 0.08 -1.56 2.43 0.45
2 1 15 10 173 0.16 -1.86 2.26 0.36
2 1 16 6 270 0.07 -0.95 1.54 0.98
2 1 17 13 108 0.08 -1.35 2.62 0.32
2 1 18 6 92 0.03 -0.96 2.69 0.52
2 1 19 8 134 0.08 -0.86 2.40 0.83
2 1 20 13 202 0.08 -1.18 2.02 0.94
2 1 21 10 323 0.06 -1.29 1.38 0.99
2 1 22 5 186 0.08 -0.92 2.07 0.96
2 1 23 9 170 0.06 -1.44 2.27 0.72
2 1 24 4 357 0.01 -1.25 1.21 0.97
2 1 25 4 136 0.07 -1.33 2.44 0.62
2 1 26 13 282 0.06 -1.14 1.52 0.99
2 1 27 7 165 0.06 -1.00 2.23 0.89
2 1 28 10 218 0.04 -0.90 1.85 1.00
2 1 29 7 95 0.07 -1.11 2.68 0.42
2 1 30 7 217 0.04 -0.69 1.79 1.00
2 2 1 17 619 0.09 1.19 -1.36 -0.97
2 2 2 17 645 0.13 1.63 -1.52 -0.97
2 2 3 16 500 0.09 0.90 -0.58 -0.38
2 2 4 14 489 0.08 2.05 -0.74 -0.98
2 2 5 11 463 0.07 1.63 -0.48 -0.95
2 2 6 9 474 0.06 1.05 -0.38 -0.46
2 2 7 8 389 0.08 1.75 -0.03 -0.97
2 2 8 11 562 0.08 1.11 -1.06 -0.88
2 2 9 10 388 0.06 1.51 0.08 -0.87
2 2 10 12 579 0.08 0.63 -1.04 -0.62
2 2 11 10 466 0.08 1.93 -0.58 -1.00
2 2 12 6 429 0.05 1.58 -0.20 -0.91
2 2 13 12 560 0.07 1.35 -1.10 -0.96
2 2 14 23 561 0.13 2.00 -1.21 -0.93
2 2 15 14 543 0.07 0.73 -0.85 -0.47
2 2 16 10 475 0.07 1.18 -0.42 -0.66
2 2 17 16 533 0.06 1.03 -0.83 -0.74
2 2 18 14 497 0.08 1.23 -0.62 -0.78
2 2 19 13 521 0.08 1.43 -0.83 -0.94
2 2 20 20 615 0.1 0.75 -1.25 -0.84
2 2 21 9 420 0.08 1.38 -0.08 -0.78
2 2 22 17 487 0.07 1.44 -0.57 -0.89
2 2 23 8 445 0.06 1.31 -0.24 -0.74
2 2 24 11 425 0.08 1.90 -0.31 -0.99
2 2 25 15 515 0.09 1.68 -0.84 -0.99
2 2 26 19 504 0.09 1.02 -0.64 -0.61
2 2 27 12 592 0.08 1.57 -1.29 -1.00
2 2 28 9 447 0.04 1.45 -0.30 -0.85
2 2 29 14 554 0.08 0.82 -0.94 -0.65
2 2 30 15 448 0.06 1.16 -0.21 -0.57
2 3 1 9 341 0.06 1.06 0.51 0.57
2 3 2 14 353 0.08 0.83 0.57 0.10
2 3 3 11 332 0.07 0.87 0.65 0.40
2 3 4 13 371 0.08 0.90 0.44 -0.10
2 3 5 9 465 0.09 1.02 -0.34 0.39
2 3 6 12 377 0.06 1.03 0.32 0.38
2 3 7 13 407 0.07 1.03 0.12 0.25
2 3 8 9 370 0.07 1.03 0.40 -0.44
2 3 9 14 406 0.06 0.99 0.15 -0.02
2 3 10 14 367 0.06 1.18 0.35 -0.64
2 3 11 13 326 0.06 0.70 0.75 0.21
2 3 12 10 436 0.08 1.01 -0.08 0.14
2 3 13 7 426 0.06 1.01 0.00 -0.16
2 3 14 14 362 0.07 0.92 0.50 -0.30
2 3 15 16 456 0.08 0.99 -0.25 0.21
2 3 16 11 434 0.06 1.12 -0.08 -0.48
2 3 17 10 457 0.07 1.00 -0.24 -0.22
2 3 18 10 472 0.08 0.94 -0.34 -0.02
2 3 19 17 363 0.08 1.17 0.33 0.62
2 3 20 13 385 0.06 0.99 0.32 -0.28
2 3 21 14 349 0.06 0.96 0.51 0.41
2 3 22 6 435 0.03 1.05 -0.08 -0.32
2 3 23 7 409 0.03 1.04 0.12 -0.31
2 3 24 12 403 0.07 1.27 0.01 0.68
2 3 25 10 432 0.09 1.11 -0.10 0.46
2 3 26 13 402 0.07 1.15 0.12 -0.54
2 3 27 14 338 0.09 1.35 0.35 0.80
2 3 28 14 378 0.07 0.94 0.36 0.12
2 3 29 17 398 0.06 1.12 0.14 0.48
2 3 30 7 372 0.05 1.31 0.25 -0.75
2 4 1 14 374 0.1 -0.92 1.03 -0.78
2 4 2 14 369 0.07 -0.69 1.00 -0.62
2 4 3 15 289 0.08 -0.86 1.41 -0.93
2 4 4 6 239 0.07 -0.19 1.53 -0.89
2 4 5 13 290 0.08 -1.11 1.47 -0.98
2 4 6 10 216 0.06 -0.66 1.80 -0.99
2 4 7 10 300 0.06 -0.37 1.24 -0.71
2 4 8 21 356 0.08 -0.38 0.98 -0.33
2 4 9 21 430 0.09 -1.31 0.86 -0.90
2 4 10 15 359 0.09 -1.44 1.28 -0.99
2 4 11 9 314 0.07 -0.24 1.14 -0.55
2 4 12 11 337 0.07 -0.47 1.11 -0.60
2 4 13 12 223 0.06 -0.49 1.69 -0.97
2 4 14 10 443 0.07 -0.86 0.59 -0.27
2 4 15 13 277 0.07 -0.51 1.38 -0.85
2 4 16 4 266 0.05 -0.16 1.36 -0.77
2 4 17 11 329 0.08 -0.68 1.19 -0.78
2 4 18 4 215 0.02 -0.14 1.65 -0.94
2 4 19 13 419 0.09 -1.57 1.01 -0.99
2 4 20 7 417 0.07 -0.72 0.73 -0.25
2 4 21 12 386 0.06 -0.61 0.90 -0.40
2 4 22 8 254 0.1 -1.25 1.71 -0.99
2 4 23 15 424 0.09 -1.07 0.79 -0.74
2 4 24 16 418 0.09 -0.83 0.77 -0.49
2 4 25 15 245 0.07 -0.92 1.69 -0.99
2 4 26 11 450 0.08 -0.96 0.58 -0.47
2 4 27 10 258 0.06 -0.65 1.54 -0.94
2 4 28 16 297 0.09 -1.52 1.54 -0.98
2 4 29 8 201 0.07 -0.33 1.78 -0.98
2 4 30 14 358 0.1 -1.02 1.16 -0.89
2 5 1 9 716 0.09 -1.02 -1.17 -0.90
2 5 2 14 837 0.14 -0.99 -2.05 -0.96
2 5 3 7 820 0.07 -1.09 -1.84 -0.99
2 5 4 3 912 0.07 -1.80 -2.26 -0.45
2 5 5 4 793 0.04 -0.77 -1.76 -1.00
2 5 6 10 675 0.08 -1.12 -0.89 -0.82
2 5 7 9 816 0.07 -2.05 -1.25 -0.91
2 5 8 14 774 0.05 -1.74 -1.16 -0.99
2 5 9 6 681 0.06 -1.27 -0.83 -0.87
2 5 10 10 770 0.09 -0.95 -1.56 -0.98
2 5 11 12 801 0.06 -1.41 -1.52 -1.00
2 5 12 12 889 0.11 -2.27 -1.75 -0.49
2 5 13 8 806 0.05 -1.76 -1.36 -0.97
2 5 14 12 862 0.11 -2.36 -1.48 -0.61
2 5 15 8 845 0.1 -2.27 -1.38 -0.75
2 5 16 16 861 0.11 -1.39 -2.06 -0.87
2 5 17 10 850 0.09 -2.00 -1.59 -0.83
2 5 18 13 895 0.12 -2.06 -1.95 -0.53
2 5 19 10 834 0.08 -1.62 -1.70 -0.93
2 5 20 9 767 0.05 -1.48 -1.26 -1.00
2 5 21 9 747 0.05 -1.91 -0.81 -1.00
2 5 22 13 817 0.09 -1.29 -1.72 -0.98
2 5 23 11 758 0.07 -1.13 -1.37 -0.97
2 5 24 13 732 0.07 -1.67 -0.89 -0.99
2 5 25 6 887 0.07 -1.55 -2.23 -0.70
2 5 26 15 743 0.07 -1.33 -1.15 -0.97
2 5 27 10 717 0.08 -1.44 -0.93 -0.96
2 5 28 13 871 0.09 -1.81 -1.92 -0.76
2 5 29 6 772 0.05 -1.94 -1.01 -0.98
2 5 30 8 811 0.08 -2.23 -1.09 -0.87
2 6 1 11 319 0.05 0.57 0.84 0.19
2 6 2 13 291 0.05 0.37 1.06 -0.48
2 6 3 10 317 0.07 0.32 0.95 -0.01
2 6 4 20 322 0.1 -0.07 1.05 -0.31
2 6 5 9 226 0.08 0.15 1.51 -0.87
2 6 6 8 312 0.06 0.34 0.96 0.21
2 6 7 14 313 0.07 0.25 1.00 -0.24
2 6 8 6 192 0.06 0.75 1.48 -0.94
2 6 9 10 179 0.08 1.10 1.43 -0.98
2 6 10 21 255 0.09 1.10 0.98 -0.85
2 6 11 17 335 0.07 0.85 0.66 -0.38
2 6 12 23 304 0.08 0.72 0.87 -0.48
2 6 13 14 324 0.07 0.05 1.00 -0.07
2 6 14 19 331 0.08 0.70 0.74 -0.18
2 6 15 16 260 0.06 0.37 1.24 -0.70
2 6 16 15 294 0.08 0.17 1.12 -0.49
2 6 17 16 301 0.08 0.93 0.79 -0.63
2 6 18 14 321 0.1 1.10 0.61 -0.67
2 6 19 15 293 0.07 0.52 1.00 -0.48
2 6 20 6 207 0.05 0.44 1.52 -0.91
2 6 21 12 275 0.11 0.00 1.27 -0.68
2 6 22 15 222 0.09 0.86 1.28 -0.89
2 6 23 15 248 0.1 0.92 1.12 -0.83
2 6 24 10 238 0.06 0.58 1.31 -0.82
2 6 25 12 318 0.04 0.22 0.98 0.12
2 6 26 10 265 0.08 0.74 1.08 -0.72
2 6 27 6 316 0.05 0.49 0.89 0.13
2 6 28 17 311 0.07 0.47 0.91 -0.23
2 6 29 12 325 0.08 0.60 0.80 -0.04
2 6 30 9 250 0.06 0.22 1.35 -0.78
2 7 1 14 786 0.08 -0.08 -2.10 0.99
2 7 2 10 618 0.08 -0.31 -1.00 0.31
2 7 3 8 627 0.04 -0.46 -0.98 0.40
2 7 4 13 701 0.08 -0.67 -1.36 0.87
2 7 5 8 629 0.06 -0.37 -1.05 0.46
2 7 6 10 651 0.06 -0.58 -1.09 0.64
2 7 7 11 630 0.06 -0.70 -0.92 0.53
2 7 8 11 637 0.08 -0.10 -1.17 0.56
2 7 9 11 740 0.08 -0.74 -1.55 0.96
2 7 10 14 807 0.08 -0.37 -2.14 0.98
2 7 11 9 726 0.07 -0.49 -1.57 0.93
2 7 12 11 683 0.08 0.24 -1.53 0.89
2 7 13 5 662 0.05 -0.18 -1.30 0.73
2 7 14 10 697 0.07 -0.06 -1.54 0.89
2 7 15 17 847 0.14 -0.45 -2.47 0.85
2 7 16 12 647 0.05 -0.29 -1.17 0.60
2 7 17 15 764 0.08 -0.50 -1.81 0.99
2 7 18 8 698 0.12 -1.15 -1.11 0.91
2 7 19 11 833 0.09 -0.72 -2.24 0.93
2 7 20 14 731 0.08 0.11 -1.80 0.98
2 7 21 10 690 0.06 -0.37 -1.37 0.81
2 7 22 4 613 0.02 -0.53 -0.90 0.30
2 7 23 14 749 0.09 -0.21 -1.78 0.97
2 7 24 15 771 0.1 -1.00 -1.67 1.00
2 7 25 11 688 0.06 -0.13 -1.44 0.83
2 7 26 10 815 0.09 -0.96 -1.97 0.98
2 7 27 17 657 0.09 -0.81 -1.03 0.72
2 7 28 14 648 0.07 0.07 -1.30 0.71
2 7 29 8 660 0.06 -0.50 -1.18 0.69
2 7 30 19 729 0.13 -1.03 -1.36 0.95
2 8 1 11 667 0.09 -1.56 -0.66 0.95
2 8 2 3 622 0.02 -1.95 -0.20 1.00
2 8 3 17 829 0.1 -2.14 -1.41 0.82
2 8 4 15 788 0.09 -2.28 -1.01 0.87
2 8 5 11 711 0.07 -1.43 -1.02 0.97
2 8 6 9 585 0.08 -1.63 -0.19 0.93
2 8 7 8 754 0.07 -2.58 -0.55 0.77
2 8 8 14 663 0.07 -1.77 -0.51 0.99
2 8 9 8 635 0.03 -1.85 -0.31 0.99
2 8 10 10 666 0.06 -2.02 -0.40 1.00
2 8 11 8 782 0.08 -1.93 -1.24 0.95
2 8 12 8 703 0.07 -1.67 -0.81 0.99
2 8 13 7 735 0.06 -1.68 -1.03 1.00
2 8 14 16 859 0.15 -2.48 -1.42 0.48
2 8 15 7 762 0.04 -1.54 -1.31 1.00
2 8 16 11 739 0.07 -1.35 -1.23 0.98
2 8 17 17 671 0.1 -1.22 -0.90 0.87
2 8 18 14 699 0.07 -2.51 -0.28 0.85
2 8 19 15 810 0.09 -2.44 -1.05 0.75
2 8 20 7 751 0.07 -2.07 -0.88 0.97
2 8 21 5 750 0.04 -1.89 -0.97 0.99
2 8 22 6 804 0.09 -2.65 -0.82 0.62
2 8 23 8 794 0.07 -1.73 -1.40 0.97
2 8 24 8 738 0.08 -2.69 -0.35 0.69
2 8 25 4 589 0.03 -1.81 -0.11 0.98
2 8 26 7 842 0.1 -2.64 -1.16 0.45
2 8 27 8 665 0.07 -2.24 -0.27 0.96
2 8 28 11 623 0.1 -1.43 -0.51 0.88
2 8 29 10 730 0.12 -2.31 -0.60 0.91
2 8 30 8 702 0.07 -2.00 -0.63 0.99
2 9 1 10 473 0.06 -1.70 0.71 -0.98
2 9 2 17 673 0.08 -2.05 -0.28 -0.99
2 9 3 20 513 0.1 -1.13 0.13 -0.50
2 9 4 16 478 0.1 -1.42 0.57 -0.88
2 9 5 14 541 0.07 -1.31 -0.01 -0.73
2 9 6 16 563 0.08 -1.98 0.21 -1.00
2 9 7 11 575 0.09 -1.06 -0.38 -0.48
2 9 8 18 520 0.08 -1.36 0.21 -0.78
2 9 9 11 510 0.08 -1.65 0.40 -0.95
2 9 10 15 546 0.08 -1.63 0.14 -0.93
2 9 11 8 616 0.06 -2.06 -0.01 -1.00
2 9 12 16 631 0.08 -1.21 -0.58 -0.75
2 9 13 19 548 0.09 -1.13 -0.17 -0.50
2 9 14 20 567 0.08 -1.52 -0.07 -0.88
2 9 15 18 481 0.1 -1.17 0.41 -0.64
2 9 16 10 708 0.06 -2.00 -0.54 -1.00
2 9 17 9 658 0.05 -1.58 -0.49 -0.94
2 9 18 16 626 0.09 -1.06 -0.64 -0.65
2 9 19 9 569 0.06 -1.24 -0.25 -0.67
2 9 20 11 661 0.05 -1.45 -0.59 -0.90
2 9 21 9 612 0.06 -1.78 -0.14 -0.98
2 9 22 20 486 0.08 -1.03 0.32 -0.40
2 9 23 9 572 0.09 -2.30 0.27 -0.95
2 9 24 15 600 0.08 -1.39 -0.34 -0.82
2 9 25 10 693 0.05 -1.75 -0.58 -0.99
2 9 26 8 638 0.03 -1.64 -0.34 -0.95
2 9 27 16 505 0.09 -1.89 0.52 -1.00
2 9 28 0 NA NA NA NA NA
2 9 29 0 NA NA NA NA NA
2 9 30 0 NA NA NA NA NA
2 10 1 11 267 0.08 1.87 0.41 0.99
2 10 2 13 227 0.11 1.73 0.74 0.99
2 10 3 7 51 0.09 2.61 1.37 0.31
2 10 4 5 118 0.05 2.92 0.68 0.02
2 10 5 8 142 0.05 1.83 1.20 0.98
2 10 6 5 141 0.08 2.73 0.66 0.58
2 10 7 7 72 0.1 2.53 1.24 0.57
2 10 8 12 219 0.08 2.10 0.58 0.98
2 10 9 8 190 0.06 1.98 0.83 0.99
2 10 10 11 110 0.08 2.79 0.85 0.39
2 10 11 12 177 0.06 1.78 1.02 1.00
2 10 12 14 53 0.1 2.71 1.27 -0.01
2 10 13 9 73 0.09 2.69 1.12 0.39
2 10 14 10 184 0.09 2.56 0.55 0.78
2 10 15 16 243 0.11 2.49 0.23 0.86
2 10 16 9 96 0.11 1.93 1.50 0.89
2 10 17 5 163 0.05 2.91 0.44 0.33
2 10 18 11 198 0.08 2.80 0.29 0.57
2 10 19 10 268 0.07 2.12 0.28 0.99
2 10 20 17 56 0.11 2.28 1.57 0.63
2 10 21 8 153 0.06 2.09 1.02 0.94
2 10 22 7 117 0.06 2.04 1.27 0.91
2 10 23 7 187 0.06 2.93 0.30 0.32
2 10 24 13 113 0.12 2.46 1.05 0.74
2 10 25 8 156 0.07 2.32 0.85 0.88
2 10 26 6 204 0.07 2.31 0.54 0.93
2 10 27 12 74 0.1 2.81 1.04 0.09
2 10 28 6 90 0.07 2.84 0.90 -0.19
2 10 29 7 138 0.05 2.93 0.56 0.18
2 10 30 15 256 0.14 2.70 0.07 0.70
2 11 1 11 607 0.07 0.40 -1.20 0.67
2 11 2 8 640 0.05 0.55 -1.39 0.86
2 11 3 9 584 0.06 0.38 -1.04 0.45
2 11 4 14 535 0.07 0.80 -0.82 0.52
2 11 5 15 484 0.06 1.02 -0.49 0.50
2 11 6 9 621 0.09 0.74 -1.33 0.88
2 11 7 15 566 0.1 0.39 -0.92 0.00
2 11 8 11 564 0.06 0.44 -0.92 0.22
2 11 9 7 605 0.06 0.21 -1.13 0.52
2 11 10 14 588 0.06 0.18 -0.98 0.00
2 11 11 13 587 0.06 0.22 -1.01 0.24
2 11 12 10 576 0.05 0.82 -1.13 0.79
2 11 13 12 551 0.06 0.59 -0.89 0.37
2 11 14 13 506 0.08 1.02 -0.72 0.66
2 11 15 15 608 0.08 -0.10 -1.03 0.27
2 11 16 10 568 0.06 0.61 -1.02 0.59
2 11 17 15 542 0.08 0.57 -0.82 0.05
2 11 18 17 582 0.08 1.04 -1.24 0.92
2 11 19 9 601 0.06 0.56 -1.21 0.75
2 11 20 18 503 0.11 0.81 -0.59 -0.04
2 11 21 12 526 0.08 0.75 -0.70 -0.22
2 11 22 15 496 0.08 0.88 -0.54 0.24
2 11 23 14 550 0.08 0.87 -0.98 0.72
2 11 24 9 511 0.07 0.86 -0.67 0.41
2 11 25 12 532 0.07 0.71 -0.73 0.20
2 11 26 5 632 0.03 0.21 -1.25 0.68
2 11 27 6 537 0.06 1.07 -0.92 0.81
2 11 28 10 603 0.04 0.06 -1.06 0.35
2 11 29 9 652 0.07 0.36 -1.40 0.83
2 11 30 30 552 0.09 0.57 -0.87 -0.28
2 12 1 6 292 0.06 -2.45 1.66 -0.28
2 12 2 12 344 0.08 -1.97 1.46 -0.89
2 12 3 7 364 0.08 -2.60 1.48 -0.11
2 12 4 9 283 0.06 -2.48 1.67 0.03
2 12 5 5 247 0.05 -2.34 1.87 0.05
2 12 6 7 455 0.09 -2.26 0.98 -0.89
2 12 7 10 211 0.08 -1.91 2.07 -0.57
2 12 8 12 237 0.09 -2.18 1.92 -0.42
2 12 9 5 387 0.05 -1.75 1.25 -0.99
2 12 10 17 470 0.12 -2.80 1.02 -0.18
2 12 11 7 309 0.08 -2.17 1.60 -0.71
2 12 12 10 444 0.12 -1.87 0.95 -0.99
2 12 13 15 393 0.09 -2.29 1.33 -0.76
2 12 14 11 498 0.08 -2.47 0.79 -0.80
2 12 15 12 528 0.13 -2.73 0.72 -0.55
2 12 16 8 274 0.08 -1.74 1.71 -0.89
2 12 17 11 530 0.12 -2.90 0.72 -0.12
2 12 18 12 307 0.11 -2.37 1.62 -0.49
2 12 19 7 452 0.09 -2.76 1.12 0.17
2 12 20 10 261 0.09 -1.93 1.79 -0.77
2 12 21 8 423 0.08 -2.12 1.12 -0.91
2 12 22 5 544 0.09 -2.57 0.52 -0.78
2 12 23 5 253 0.07 -2.32 1.82 0.31
2 12 24 10 519 0.06 -2.30 0.60 -0.93
2 12 25 14 354 0.11 -2.55 1.49 0.29
2 12 26 6 259 0.08 -2.39 1.79 -0.17
2 12 27 13 400 0.09 -2.68 1.34 0.04
2 12 28 8 209 0.08 -2.18 2.05 -0.01
2 12 29 20 439 0.12 -2.60 1.17 -0.52
2 12 30 5 183 0.08 -2.00 2.19 -0.24
2 13 1 9 85 0.09 2.57 1.11 -0.59
2 13 2 14 149 0.09 1.37 1.47 -0.99
2 13 3 8 157 0.06 2.14 0.94 -0.94
2 13 4 20 97 0.1 1.67 1.62 -0.94
2 13 5 13 49 0.09 2.53 1.43 -0.41
2 13 6 9 169 0.08 2.44 0.67 -0.84
2 13 7 18 285 0.11 1.32 0.65 -0.85
2 13 8 9 220 0.08 1.31 1.07 -0.95
2 13 9 8 67 0.06 2.42 1.32 -0.65
2 13 10 9 30 0.11 2.31 1.75 -0.43
2 13 11 11 88 0.1 2.76 0.98 -0.36
2 13 12 18 246 0.09 1.48 0.82 -0.95
2 13 13 14 124 0.12 1.99 1.24 -0.93
2 13 14 16 272 0.08 1.69 0.53 -0.97
2 13 15 11 154 0.08 2.69 0.62 -0.64
2 13 16 15 196 0.07 1.88 0.85 -1.00
2 13 17 12 333 0.1 1.55 0.31 -0.91
2 13 18 8 87 0.07 2.16 1.38 -0.82
2 13 19 8 197 0.09 2.10 0.70 -0.97
2 13 20 17 230 0.08 2.29 0.41 -0.94
2 13 21 6 50 0.09 2.16 1.64 -0.69
2 13 22 14 125 0.12 2.41 0.97 -0.80
2 13 23 10 279 0.07 2.13 0.22 -0.99
2 13 24 9 340 0.08 1.85 0.14 -0.99
2 13 25 14 42 0.13 1.93 1.86 -0.71
2 13 26 5 45 0.1 2.63 1.38 -0.21
2 13 27 6 206 0.08 2.49 0.43 -0.85
2 13 28 10 264 0.08 1.88 0.44 -0.99
2 13 29 13 178 0.11 1.59 1.16 -1.00
2 13 30 9 103 0.08 2.29 1.21 -0.81
2 14 1 6 234 0.08 2.99 0.02 -0.05
2 14 2 9 516 0.07 2.76 -1.12 -0.21
2 14 3 18 306 0.12 2.60 -0.11 -0.79
2 14 4 7 431 0.06 2.90 -0.69 0.18
2 14 5 7 461 0.11 2.87 -0.86 -0.06
2 14 6 9 525 0.06 2.45 -1.06 -0.74
2 14 7 16 241 0.12 2.62 0.19 -0.77
2 14 8 3 233 0.06 2.94 0.07 0.32
2 14 9 5 280 0.05 2.94 -0.19 0.33
2 14 10 9 139 0.07 2.92 0.55 -0.21
2 14 11 4 167 0.06 2.97 0.36 0.04
2 14 12 9 438 0.08 2.80 -0.70 0.46
2 14 13 8 468 0.06 2.82 -0.89 0.27
2 14 14 7 181 0.07 2.95 0.32 -0.25
2 14 15 20 276 0.09 2.90 -0.14 -0.42
2 14 16 10 373 0.12 2.20 -0.16 -0.97
2 14 17 8 383 0.08 2.93 -0.50 0.24
2 14 18 11 483 0.08 2.59 -0.84 -0.69
2 14 19 13 480 0.08 2.36 -0.79 -0.87
2 14 20 11 288 0.08 2.34 0.06 -0.94
2 14 21 7 509 0.09 2.66 -1.08 -0.48
2 14 22 5 252 0.01 2.98 -0.06 0.19
2 14 23 7 404 0.1 2.64 -0.45 -0.73
2 14 24 5 336 0.09 2.87 -0.32 0.45
2 14 25 10 390 0.12 2.88 -0.48 -0.38
2 14 26 10 437 0.09 2.33 -0.52 -0.92
2 14 27 10 166 0.07 2.83 0.44 -0.50
2 14 28 10 232 0.12 2.88 0.10 -0.46
2 14 29 10 454 0.09 2.81 -0.77 -0.40
2 14 30 11 330 0.09 2.97 -0.35 -0.05
2 15 1 11 78 0.06 1.04 2.09 -0.94
2 15 2 12 68 0.1 1.34 2.00 -0.91
2 15 3 15 39 0.13 0.63 2.64 -0.69
2 15 4 5 185 0.06 0.44 1.66 -0.96
2 15 5 6 13 0.1 0.56 2.94 -0.04
2 15 6 13 10 0.1 0.89 2.84 -0.20
2 15 7 7 34 0.05 1.43 2.30 -0.70
2 15 8 11 62 0.09 0.42 2.51 -0.84
2 15 9 7 18 0.1 1.22 2.58 -0.51
2 15 10 13 143 0.06 0.97 1.68 -1.00
2 15 11 15 43 0.14 0.17 2.75 -0.65
2 15 12 10 63 0.1 0.91 2.30 -0.88
2 15 13 9 194 0.06 0.04 1.71 -0.96
2 15 14 12 36 0.09 1.21 2.40 -0.72
2 15 15 16 27 0.14 0.11 2.95 -0.29
2 15 16 6 111 0.07 0.96 1.92 -0.99
2 15 17 10 145 0.06 0.32 1.91 -0.99
2 15 18 5 91 0.08 -0.11 2.42 -0.90
2 15 19 13 64 0.07 0.02 2.62 -0.78
2 15 20 7 99 0.05 0.37 2.22 -0.97
2 15 21 6 105 0.06 1.15 1.86 -0.98
2 15 22 11 147 0.07 0.10 1.98 -1.00
2 15 23 8 86 0.07 0.63 2.22 -0.95
2 15 24 10 22 0.09 0.42 2.89 -0.37
2 15 25 8 132 0.06 0.56 1.91 -1.00
2 15 26 7 155 0.07 -0.19 2.02 -1.00
2 15 27 11 164 0.07 0.74 1.66 -0.98
2 15 28 8 120 0.06 0.04 2.16 -0.99
2 15 29 9 94 0.06 0.04 2.35 -0.93
2 15 30 11 24 0.11 0.96 2.65 -0.57
2 16 1 8 40 0.09 -0.21 2.92 -0.37
2 16 2 9 208 0.05 -1.23 1.99 -0.94
2 16 3 13 152 0.12 -1.78 2.34 -0.31
2 16 4 5 146 0.07 -1.19 2.37 -0.76
2 16 5 8 107 0.09 -1.12 2.61 -0.54
2 16 6 8 48 0.05 -0.57 2.94 0.05
2 16 7 12 60 0.11 -0.29 2.75 -0.63
2 16 8 8 148 0.11 -0.37 2.14 -0.98
2 16 9 8 109 0.1 -1.51 2.59 -0.03
2 16 10 7 61 0.08 -0.64 2.84 -0.41
2 16 11 6 225 0.06 -1.46 1.94 -0.90
2 16 12 12 76 0.08 -0.57 2.70 -0.65
2 16 13 8 162 0.1 -1.56 2.28 -0.64
2 16 14 7 188 0.08 -1.49 2.15 -0.79
2 16 15 6 47 0.07 -0.51 2.93 -0.22
2 16 16 4 210 0.03 -1.67 2.06 -0.75
2 16 17 9 93 0.09 -0.89 2.67 -0.58
2 16 18 7 65 0.09 -0.87 2.86 0.13
2 16 19 10 130 0.08 -1.53 2.45 -0.45
2 16 20 16 140 0.07 -0.92 2.32 -0.87

Now let’s check the compression summary for HVT (torus_mapC). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- torus_mapC[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 30 0 0 n_cells: 30 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 894 767 0.86 n_cells: 30 quant.err: 0.1 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 86% of the cells have hit the quantization threshold error in level 2

Let’s plot the Voronoi tessellation for layer 2 (map C)

plotHVT(torus_mapC,
        line.width = c(0.4,0.2), 
        color.vec = c("#141B41","#0582CA"),
        centroid.size = 0.1,
        maxDepth = 2, 
        heatmap = '2Dhvt') 
Figure 14: The Voronoi Tessellation for layer 2 (map C) shown for the 924 cells in the dataset ’torus’ at level 2

Figure 14: The Voronoi Tessellation for layer 2 (map C) shown for the 924 cells in the dataset ’torus’ at level 2

Heat Maps

Now let’s plot all the features for each cell at level two as a heatmap for better visualization.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the torus data, allowing us to observe patterns and trends in the distribution of each of the features (x,y,z). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the torus data.


  plotHVT(
  torus_mapC,
  trainTorus_data,
  child.level = 2,
  hmap.cols = "x",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
  heatmap = '2Dheatmap'
) 
Figure 15: The Voronoi Tessellation with the heat map overlaid for feature `x` in the ’torus’ dataset

Figure 15: The Voronoi Tessellation with the heat map overlaid for feature x in the ’torus’ dataset


  plotHVT(
  torus_mapC,
  trainTorus_data,
  child.level = 2,
  hmap.cols = "y",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
  heatmap = '2Dheatmap'
) 
Figure 16: The Voronoi Tessellation with the heat map overlaid for feature `y` in the ’torus’ dataset

Figure 16: The Voronoi Tessellation with the heat map overlaid for feature y in the ’torus’ dataset


  plotHVT(
  torus_mapC,
  trainTorus_data,
  child.level = 2,
  hmap.cols = "z",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = TRUE,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
  heatmap = '2Dheatmap'
) 
Figure 17: The Voronoi Tessellation with the heat map overlaid for feature `z` in the ’torus’ dataset

Figure 17: The Voronoi Tessellation with the heat map overlaid for feature z in the ’torus’ dataset

We now have the set of maps (map A, map B & map C) which will be used to score, which map and cell each test record is assigned to, but before that lets view our test dataset

6. Scoring on Test Data

Now once we have built the model, let us try to score using our test dataset (containing 2400 data points) which cell and which layer each point belongs to.

Testing Dataset

The testing dataset includes the following columns:

Let’s have a look at our randomly selected test dataset containing 2400 datapoints.

Table(head(testTorus_data))
x y z
-1.8031 1.5092 0.9362
1.1817 -1.0655 -0.9126
0.9942 -1.2500 -0.9153
1.0669 -0.0514 -0.3627
0.5570 -0.8837 -0.2954
0.8776 -0.4958 -0.1259

The scoreLayeredHVT function is used to score the test data using the scored set of maps. This function takes an input - a test data and a set of maps (map A, map B, map C).

Now, Let us understand the scoreLayeredHVT function.

scoreLayeredHVT(data,
                map_A,
                map_B,
                map_C,
                mad.threshold = 0.2,
                normalize = TRUE, 
                distance_metric="L1_Norm",
                error_metric="max",
                child.level = 1, 
                line.width = c(0.6, 0.4, 0.2),
                color.vec = c("#141B41", "#6369D1", "#D8D2E1"),
                yVar= NULL,
                ...)

Each of the parameters of scoreLayeredHVT function has been explained below:

The function score based on the HVT maps - map A, map B and map C, constructed using trainHVT function. For each test record, the function will assign that record to Layer1 or Layer2. Layer1 contains the cell ids from map A and Layer 2 contains cell ids from map B (novelty map) and map C (map without novelty).

Scoring Algorithm

The Scoring algorithm recursively calculates the distance between each point in the test dataset and the cell centroids for each level. The following steps explain the scoring method for a single point in the test dataset:

  1. Calculate the distance between the point and the centroid of all the cells in the first level.
  2. Find the cell whose centroid has minimum distance to the point.
  3. Check if the cell drills down further to form more cells.
  4. If it doesn’t, return the path. Or else repeat steps 1 to 4 till we reach a level at which the cell doesn’t drill down further.

Note : The Scoring algorithm will not work if some of the variables used to perform quantization are missing. In the test dataset, we should not remove any features


validation_data <- testTorus_data
new_score <- scoreLayeredHVT(
    data=validation_data,
    torus_mapA,
    torus_mapB,
    torus_mapC,
    normalize = FALSE
  )

Let’s see which cell and layer each point belongs to and check the Mean Absolute Difference for each of the 2400 records.


act_pred <- new_score[["actual_predictedTable"]]
rownames(act_pred) <- NULL
act_pred %>% head(1000) %>%as.data.frame() %>%Table(scroll = TRUE)
Row.Number act_x act_y act_z Layer1.Cell.ID Layer2.Cell.ID pred_x pred_y pred_z diff
1 -1.8031 1.5092 0.9362 A115 C224 -1.4293520 1.9746821 0.7526839 0.3409154
2 1.1817 -1.0655 -0.9126 A670 C508 1.3298237 -0.7586967 -0.8086783 0.1862829
3 0.9942 -1.2500 -0.9153 A637 C508 1.3298237 -0.7586967 -0.8086783 0.3111829
4 1.0669 -0.0514 -0.3627 A629 C397 1.0336915 0.2140921 0.0921008 0.2511671
5 0.5570 -0.8837 -0.2954 A560 C556 0.6025305 -0.9291853 0.3583787 0.2482649
6 0.8776 -0.4958 -0.1259 A601 C397 1.0336915 0.2140921 0.0921008 0.3613281
7 -2.3233 -0.7173 0.9021 A93 C728 -1.9816416 -0.8223302 0.8717976 0.1589970
8 -1.4379 -1.4182 -0.9998 A201 C802 -1.5935443 -1.4685237 -0.8727753 0.1109976
9 -1.1262 0.4838 -0.6328 A286 C559 -1.4876296 -0.0265439 -0.7952792 0.3447509
10 -0.1641 0.9975 -0.1472 A422 C350 -0.5144181 1.0433207 0.4973778 0.3469055
11 0.9033 -1.7767 1.0000 A705 C763 0.7912866 -2.2489217 0.7766609 0.2691914
12 2.2775 -0.0413 -0.9606 A815 C368 2.7225869 -0.3304495 -0.3889044 0.4353107
13 1.0593 -0.8066 -0.7437 A652 C508 1.3298237 -0.7586967 -0.8086783 0.1278018
14 1.4834 0.4428 0.8920 A685 C459 1.7910516 -0.5739146 0.9028013 0.4450558
15 0.2045 1.1079 -0.4871 A479 C287 0.5468719 1.0211322 -0.4670886 0.1497170
16 2.2440 -1.5745 -0.6712 A873 C676 2.1950559 -1.7495404 0.0972307 0.3308051
17 -0.9745 2.8003 0.2623 A192 C123 -0.9422671 2.4806302 -0.5791599 0.3977876
18 0.5850 0.8331 -0.1890 A532 C287 0.5468719 1.0211322 -0.4670886 0.1680830
19 -1.5044 2.3446 0.6185 A135 C224 -1.4293520 1.9746821 0.7526839 0.1930499
20 1.5809 -0.4652 0.9360 A748 C459 1.7910516 -0.5739146 0.9028013 0.1173550
21 -1.9374 -2.2744 -0.1564 A109 C873 -1.6544946 -2.1236132 0.5230898 0.3710607
22 -0.8860 0.6243 0.4009 A333 C350 -0.5144181 1.0433207 0.4973778 0.2956935
23 0.9783 2.2893 -0.8720 A674 C81 0.5783356 2.3154644 -0.7661384 0.1773301
24 0.8406 2.1602 -0.9481 A674 C81 0.5783356 2.3154644 -0.7661384 0.1998301
25 0.4730 2.9532 0.1347 A558 C100 -0.0270481 2.4238631 0.7389638 0.5445496
26 -0.5176 -0.8570 -0.0496 A393 C655 -0.2597926 -1.1730025 -0.5212327 0.3484809
27 2.4540 -1.0923 -0.7275 A854 C368 2.7225869 -0.3304495 -0.3889044 0.4563444
28 2.2682 1.9181 -0.2412 A848 B1 2.2784556 1.9447222 0.0217667 0.0999481
29 2.5442 -1.4977 -0.3052 A882 C676 2.1950559 -1.7495404 0.0972307 0.3344717
30 0.0410 1.0727 -0.3763 A479 C287 0.5468719 1.0211322 -0.4670886 0.2160761
31 -0.8827 -0.7320 -0.5215 A359 C655 -0.2597926 -1.1730025 -0.5212327 0.3547258
32 -2.2024 -0.7311 0.9472 A93 C728 -1.9816416 -0.8223302 0.8717976 0.1291304
33 -2.3576 1.3298 0.7075 A40 C491 -2.1349643 0.6623576 0.8377438 0.3401073
34 0.9179 0.7176 -0.5504 A608 C287 0.5468719 1.0211322 -0.4670886 0.2526239
35 0.4195 -0.9159 0.1218 A525 C556 0.6025305 -0.9291853 0.3583787 0.1442982
36 1.5297 1.9185 -0.8912 A723 C175 1.9954496 0.9735682 -0.8296280 0.4907511
37 -0.1005 -0.9954 -0.0290 A448 C655 -0.2597926 -1.1730025 -0.5212327 0.2763759
38 1.7796 0.1579 -0.9770 A749 C175 1.9954496 0.9735682 -0.8296280 0.3929633
39 2.0152 1.8853 0.6504 A803 C29 1.5896815 2.2890353 0.3101313 0.3898408
40 -0.8793 -1.8737 0.9976 A269 C710 -0.4532110 -1.5200102 0.8113151 0.3220212
41 0.4925 -2.4521 -0.8654 A673 C776 0.8378659 -2.2297098 -0.7821201 0.2170120
42 1.5076 -1.4888 -0.9929 A768 C508 1.3298237 -0.7586967 -0.8086783 0.3640338
43 0.9571 2.7347 0.4413 A700 C29 1.5896815 2.2890353 0.3101313 0.4031383
44 1.6536 -0.9304 -0.9947 A751 C508 1.3298237 -0.7586967 -0.8086783 0.2271671
45 1.7288 -0.7202 0.9919 A748 C459 1.7910516 -0.5739146 0.9028013 0.0992119
46 -1.3049 -0.9790 -0.9296 A230 C802 -1.5935443 -1.4685237 -0.8727753 0.2783309
47 0.3303 2.4880 0.8603 A488 C100 -0.0270481 2.4238631 0.7389638 0.1809404
48 -0.9228 -2.5717 0.6811 A340 C873 -1.6544946 -2.1236132 0.5230898 0.4459306
49 2.4008 0.4820 -0.8937 A802 C175 1.9954496 0.9735682 -0.8296280 0.3203302
50 1.2205 -2.0992 -0.9037 A740 C776 0.8378659 -2.2297098 -0.7821201 0.2115746
51 2.5423 0.7690 0.7547 A826 C160 2.3972612 0.8131370 0.6512993 0.0975255
52 2.7894 -0.6793 -0.4915 A888 C368 2.7225869 -0.3304495 -0.3889044 0.1727531
53 -1.5250 2.5695 0.1549 A99 C224 -1.4293520 1.9746821 0.7526839 0.4294166
54 1.1750 1.3937 0.9842 A635 C228 0.8994003 1.2236527 0.8271905 0.2008855
55 -2.1343 -1.4370 0.8195 A108 C728 -1.9816416 -0.8223302 0.8717976 0.2732086
56 0.4443 -1.0409 -0.4962 A542 C655 -0.2597926 -1.1730025 -0.5212327 0.2870759
57 -1.4290 0.1447 0.8260 A227 C536 -1.0323493 -0.1417128 0.3010375 0.4026754
58 -1.1483 0.6003 0.7099 A258 C491 -2.1349643 0.6623576 0.8377438 0.3921886
59 0.6935 1.5007 -0.9379 A582 C287 0.5468719 1.0211322 -0.4670886 0.3656691
60 -1.6762 0.3047 -0.9551 A182 C559 -1.4876296 -0.0265439 -0.7952792 0.2265450
61 1.2452 0.5386 -0.7657 A662 C175 1.9954496 0.9735682 -0.8296280 0.4163819
62 -0.8474 -1.8388 -0.9997 A377 C802 -1.5935443 -1.4685237 -0.8727753 0.4144484
63 -0.1779 -1.1912 -0.6058 A452 C655 -0.2597926 -1.1730025 -0.5212327 0.0615525
64 -1.8148 -1.3607 -0.9634 A165 C802 -1.5935443 -1.4685237 -0.8727753 0.1399013
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541 -1.7761 1.8648 0.8180 A117 C224 -1.4293520 1.9746821 0.7526839 0.1739821
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673 0.9326 -0.4126 -0.1979 A601 C397 1.0336915 0.2140921 0.0921008 0.3392615
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675 2.3383 -1.8533 0.1802 A897 C676 2.1950559 -1.7495404 0.0972307 0.1099910
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684 2.1898 -1.2694 -0.8473 A821 C508 1.3298237 -0.7586967 -0.8086783 0.4697671
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687 0.8323 1.6587 0.9896 A572 C228 0.8994003 1.2236527 0.8271905 0.2215190
688 1.5240 2.3062 -0.6449 A752 C29 1.5896815 2.2890353 0.3101313 0.3459592
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690 0.8780 -1.7413 -0.9988 A707 C776 0.8378659 -2.2297098 -0.7821201 0.2484080
691 -1.0288 -1.9140 0.9849 A269 C710 -0.4532110 -1.5200102 0.8113151 0.3810546
692 2.7582 -1.0012 -0.3565 A895 C368 2.7225869 -0.3304495 -0.3889044 0.2462560
693 -1.2394 2.6797 0.3049 A137 C224 -1.4293520 1.9746821 0.7526839 0.4475846
694 -0.7087 -1.0217 -0.6539 A394 C655 -0.2597926 -1.1730025 -0.5212327 0.2442924
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696 0.2676 -2.7273 0.6722 A617 C763 0.7912866 -2.2489217 0.7766609 0.3688419
697 1.2457 2.4540 0.6591 A716 C29 1.5896815 2.2890353 0.3101313 0.2859716
698 -0.8476 1.6599 0.9907 A254 C224 -1.4293520 1.9746821 0.7526839 0.3781834
699 -2.7661 -0.9126 -0.4085 A33 C766 -2.7284759 -0.4976307 -0.3415241 0.1731898
700 2.6008 1.3228 0.3969 A858 C160 2.3972612 0.8131370 0.6512993 0.3225337
701 0.7697 -0.8689 0.5438 A584 C556 0.6025305 -0.9291853 0.3583787 0.1376253
702 0.5893 1.1522 -0.7084 A562 C287 0.5468719 1.0211322 -0.4670886 0.1382691
703 -0.5819 1.7348 0.9854 A332 C350 -0.5144181 1.0433207 0.4973778 0.4156611
704 0.1829 1.3122 0.7377 A461 C228 0.8994003 1.2236527 0.8271905 0.2981794
705 2.4055 -1.7710 0.1599 A897 C676 2.1950559 -1.7495404 0.0972307 0.0981910
706 1.6999 2.4717 -0.0212 A801 C29 1.5896815 2.2890353 0.3101313 0.2080715
707 0.3959 -1.6483 -0.9524 A555 C776 0.8378659 -2.2297098 -0.7821201 0.3978852
708 -1.9215 -1.9377 0.6846 A91 C873 -1.6544946 -2.1236132 0.5230898 0.2048096
709 -0.9732 2.7275 0.4442 A197 C123 -0.9422671 2.4806302 -0.5791599 0.4337209
710 1.5287 0.4109 0.9089 A685 C459 1.7910516 -0.5739146 0.9028013 0.4177550
711 1.0976 2.6824 0.4394 A700 C29 1.5896815 2.2890353 0.3101313 0.3382383
712 0.8678 0.5047 0.0878 A575 C397 1.0336915 0.2140921 0.0921008 0.1536001
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714 1.1273 -0.0050 0.4883 A624 C397 1.0336915 0.2140921 0.0921008 0.2362999
715 1.8537 -0.2190 -0.9911 A791 C508 1.3298237 -0.7586967 -0.8086783 0.4153315
716 -1.6331 0.9085 0.9914 A155 C491 -2.1349643 0.6623576 0.8377438 0.3005543
717 2.9290 0.6330 -0.0821 A871 C160 2.3972612 0.8131370 0.6512993 0.4817584
718 -2.4030 -0.7364 0.8582 A86 C728 -1.9816416 -0.8223302 0.8717976 0.1736288
719 1.4092 -0.3997 -0.8447 A701 C508 1.3298237 -0.7586967 -0.8086783 0.1581315
720 0.3504 1.0666 0.4800 A492 C350 -0.5144181 1.0433207 0.4973778 0.3018251
721 1.7803 -0.3913 -0.9842 A761 C508 1.3298237 -0.7586967 -0.8086783 0.3311315
722 0.5181 0.9235 0.3381 A514 C287 0.5468719 1.0211322 -0.4670886 0.3105309
723 -1.3025 1.7994 0.9752 A152 C224 -1.4293520 1.9746821 0.7526839 0.1748834
724 -0.0468 -1.3154 -0.7297 A480 C655 -0.2597926 -1.1730025 -0.5212327 0.1879525
725 1.8053 -1.1500 -0.9901 A784 C508 1.3298237 -0.7586967 -0.8086783 0.3494004
726 0.9214 1.7996 0.9998 A653 C228 0.8994003 1.2236527 0.8271905 0.2568521
727 2.8567 0.4996 0.4358 A875 C160 2.3972612 0.8131370 0.6512993 0.3294917
728 1.3966 0.3745 0.8325 A685 C397 1.0336915 0.2140921 0.0921008 0.4212385
729 0.7356 -0.7194 -0.2385 A560 C556 0.6025305 -0.9291853 0.3583787 0.3132445
730 1.4971 0.9398 0.9726 A696 C228 0.8994003 1.2236527 0.8271905 0.3423206
731 -0.6036 -2.7538 0.5735 A429 C881 -0.5693053 -2.7024202 -0.2819962 0.3137235
732 1.0188 -0.8862 -0.7602 A652 C508 1.3298237 -0.7586967 -0.8086783 0.1623351
733 2.9457 -0.5592 -0.0587 A891 C368 2.7225869 -0.3304495 -0.3889044 0.2606893
734 -1.3756 -1.2583 0.9907 A229 C728 -1.9816416 -0.8223302 0.8717976 0.3869712
735 2.5068 -1.2010 0.6263 A866 C676 2.1950559 -1.7495404 0.0972307 0.4631179
736 0.4084 2.4953 -0.8489 A526 C81 0.5783356 2.3154644 -0.7661384 0.1441776
737 -1.3132 -0.1509 -0.7350 A247 C559 -1.4876296 -0.0265439 -0.7952792 0.1196883
738 0.8587 -1.8953 0.9967 A705 C763 0.7912866 -2.2489217 0.7766609 0.2136914
739 -1.0051 -0.8628 0.7375 A285 C728 -1.9816416 -0.8223302 0.8717976 0.3837696
740 1.0835 -0.8138 -0.7642 A652 C508 1.3298237 -0.7586967 -0.8086783 0.1153018
741 -1.1953 -1.7418 -0.9937 A244 C802 -1.5935443 -1.4685237 -0.8727753 0.2641484
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743 -0.0005 2.0683 0.9977 A444 C100 -0.0270481 2.4238631 0.7389638 0.2136158
744 1.3984 -1.3943 -0.9997 A738 C508 1.3298237 -0.7586967 -0.8086783 0.2984004
745 1.8194 2.3066 0.3472 A801 C29 1.5896815 2.2890353 0.3101313 0.0947839
746 -0.9696 0.2533 -0.0662 A315 C536 -1.0323493 -0.1417128 0.3010375 0.2749999
747 -0.6299 0.7788 0.0579 A379 C350 -0.5144181 1.0433207 0.4973778 0.2731601
748 1.7060 -2.3834 0.3648 A860 C676 2.1950559 -1.7495404 0.0972307 0.4634949
749 1.5026 -0.6769 0.9360 A709 C459 1.7910516 -0.5739146 0.9028013 0.1415452
750 0.5832 -2.1223 0.9796 A609 C763 0.7912866 -2.2489217 0.7766609 0.1792158
751 0.7861 1.0137 0.6968 A570 C228 0.8994003 1.2236527 0.8271905 0.1512145
752 -1.9767 -1.3728 -0.9136 A124 C802 -1.5935443 -1.4685237 -0.8727753 0.1732347
753 -1.5887 -0.3799 -0.9304 A198 C559 -1.4876296 -0.0265439 -0.7952792 0.1965157
754 -0.6474 1.1225 0.7101 A361 C350 -0.5144181 1.0433207 0.4973778 0.1416278
755 2.9618 -0.0985 0.2681 A884 C368 2.7225869 -0.3304495 -0.3889044 0.3760556
756 0.1145 -1.5462 -0.8932 A476 C655 -0.2597926 -1.1730025 -0.5212327 0.3731525
757 1.3199 0.3591 0.7748 A648 C397 1.0336915 0.2140921 0.0921008 0.3713052
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759 -0.3169 -2.9822 0.0443 A471 C881 -0.5693053 -2.7024202 -0.2819962 0.2861604
760 -0.4887 1.8193 0.9932 A338 C350 -0.5144181 1.0433207 0.4973778 0.4325065
761 2.8001 -0.5224 -0.5294 A869 C368 2.7225869 -0.3304495 -0.3889044 0.1366531
762 2.8450 -0.8996 0.1792 A892 C368 2.7225869 -0.3304495 -0.3889044 0.4198893
763 1.0436 0.4599 -0.5111 A616 C397 1.0336915 0.2140921 0.0921008 0.2863058
764 0.5061 -1.1539 -0.6726 A542 C655 -0.2597926 -1.1730025 -0.5212327 0.3121208
765 -0.8963 0.7848 0.5883 A293 C350 -0.5144181 1.0433207 0.4973778 0.2437749
766 2.4513 -1.1179 -0.7198 A854 C368 2.7225869 -0.3304495 -0.3889044 0.4632110
767 1.7514 2.4179 0.1694 A801 C29 1.5896815 2.2890353 0.3101313 0.1437715
768 -0.3954 2.8410 -0.4960 A353 C123 -0.9422671 2.4806302 -0.5791599 0.3301323
769 0.4751 -2.1536 -0.9787 A667 C776 0.8378659 -2.2297098 -0.7821201 0.2118186
770 -2.0393 -1.2246 -0.9255 A124 C802 -1.5935443 -1.4685237 -0.8727753 0.2474680
771 2.4005 -0.0915 -0.9155 A815 C368 2.7225869 -0.3304495 -0.3889044 0.3625440
772 -2.3548 -0.9765 0.8357 A86 C728 -1.9816416 -0.8223302 0.8717976 0.1878086
773 1.2128 2.5475 0.5703 A700 C29 1.5896815 2.2890353 0.3101313 0.2985050
774 0.5125 0.8649 -0.1029 A532 C287 0.5468719 1.0211322 -0.4670886 0.1849309
775 -1.4884 -0.3639 -0.8839 A236 C559 -1.4876296 -0.0265439 -0.7952792 0.1422491
776 1.2618 1.2080 0.9674 A683 C228 0.8994003 1.2236527 0.8271905 0.1727540
777 2.4297 1.2723 0.6697 A829 C160 2.3972612 0.8131370 0.6512993 0.1700008
778 -0.3028 1.6778 0.9555 A387 C100 -0.0270481 2.4238631 0.7389638 0.4127837
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780 -2.1308 1.9345 -0.4788 A36 C391 -2.3605439 1.3502882 -0.4417122 0.2836812
781 1.1060 0.2686 -0.5072 A633 C397 1.0336915 0.2140921 0.0921008 0.2420391
782 -2.0614 0.4863 -0.9930 A148 C559 -1.4876296 -0.0265439 -0.7952792 0.4281117
783 -2.1095 -0.7776 -0.9687 A151 C802 -1.5935443 -1.4685237 -0.8727753 0.4342680
784 1.0880 0.4100 0.5466 A619 C397 1.0336915 0.2140921 0.0921008 0.2349052
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786 -1.3837 2.1356 -0.8387 A179 C123 -0.9422671 2.4806302 -0.5791599 0.3486677
787 -1.9571 -2.2288 0.2581 A76 C873 -1.6544946 -2.1236132 0.5230898 0.2242607
788 -0.1386 2.3071 -0.9503 A450 C81 0.5783356 2.3154644 -0.7661384 0.3031539
789 1.9489 1.6841 -0.8177 A764 C175 1.9954496 0.9735682 -0.8296280 0.2563364
790 1.2229 -1.5158 0.9986 A730 C763 0.7912866 -2.2489217 0.7766609 0.4622247
791 -1.2651 1.0873 -0.9433 A242 C342 -0.8373696 1.1920791 -0.7640059 0.2372679
792 1.3984 2.5134 0.4820 A766 C29 1.5896815 2.2890353 0.3101313 0.1958383
793 0.4248 -2.9574 0.1560 A666 C881 -0.5693053 -2.7024202 -0.2819962 0.5623604
794 -0.6322 -0.7750 -0.0190 A393 C655 -0.2597926 -1.1730025 -0.5212327 0.4242142
795 -2.5513 -0.1628 -0.8308 A87 C766 -2.7284759 -0.4976307 -0.3415241 0.3337608
796 -1.4412 -2.5578 0.3522 A157 C873 -1.6544946 -2.1236132 0.5230898 0.2727904
797 2.7424 -1.1609 -0.2087 A895 C368 2.7225869 -0.3304495 -0.3889044 0.3434893
798 0.8014 -0.6374 -0.2178 A599 C556 0.6025305 -0.9291853 0.3583787 0.3556112
799 1.2973 -1.7640 -0.9818 A780 C776 0.8378659 -2.2297098 -0.7821201 0.3749413
800 1.0011 0.2249 0.2269 A606 C397 1.0336915 0.2140921 0.0921008 0.0593995
801 0.7159 -0.8662 0.4819 A584 C556 0.6025305 -0.9291853 0.3583787 0.0999587
802 0.8922 -0.5875 0.3630 A600 C556 0.6025305 -0.9291853 0.3583787 0.2119920
803 -1.9785 -0.0361 -0.9998 A139 C559 -1.4876296 -0.0265439 -0.7952792 0.2349824
804 -1.3674 2.1201 -0.8524 A179 C123 -0.9422671 2.4806302 -0.5791599 0.3529677
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806 1.2211 -2.0825 -0.9102 A740 C776 0.8378659 -2.2297098 -0.7821201 0.2195080
807 0.5222 2.2603 0.9475 A547 C100 -0.0270481 2.4238631 0.7389638 0.3071158
808 2.7256 1.2202 -0.1652 A859 C160 2.3972612 0.8131370 0.6512993 0.5173003
809 -1.3923 -1.2367 -0.9905 A208 C802 -1.5935443 -1.4685237 -0.8727753 0.1835976
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812 1.8120 -2.3875 0.0744 A863 B3 1.7531600 -2.4276200 -0.0842200 0.0858600
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814 1.3049 1.7304 -0.9859 A726 C81 0.5783356 2.3154644 -0.7661384 0.5104635
815 -1.5047 1.4938 -0.9927 A167 C342 -0.8373696 1.1920791 -0.7640059 0.3992485
816 1.6961 -1.2147 -0.9963 A784 C508 1.3298237 -0.7586967 -0.8086783 0.3366338
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821 -1.3379 2.6052 0.3710 A121 C224 -1.4293520 1.9746821 0.7526839 0.3678846
822 -1.3311 -0.4546 -0.8049 A236 C559 -1.4876296 -0.0265439 -0.7952792 0.1980688
823 2.5796 0.8393 0.7014 A829 C160 2.3972612 0.8131370 0.6512993 0.0862008
824 1.0925 -0.5470 0.6280 A644 C459 1.7910516 -0.5739146 0.9028013 0.3334225
825 0.0623 1.5983 0.9163 A439 C100 -0.0270481 2.4238631 0.7389638 0.3640825
826 -2.8267 0.7702 -0.3682 A25 C391 -2.3605439 1.3502882 -0.4417122 0.3732521
827 1.2564 1.9053 0.9593 A694 C228 0.8994003 1.2236527 0.8271905 0.3902521
828 1.1981 2.2696 -0.8241 A729 C81 0.5783356 2.3154644 -0.7661384 0.2411968
829 -1.7416 1.4897 0.9565 A146 C224 -1.4293520 1.9746821 0.7526839 0.3336821
830 1.0336 -0.2709 -0.3639 A625 C397 1.0336915 0.2140921 0.0921008 0.3136948
831 0.4195 2.2638 -0.9532 A526 C81 0.5783356 2.3154644 -0.7661384 0.1325205
832 1.2244 0.8993 0.8768 A650 C228 0.8994003 1.2236527 0.8271905 0.2329873
833 -0.1695 0.9855 -0.0037 A426 C350 -0.5144181 1.0433207 0.4973778 0.3012722
834 -2.1325 1.0724 0.9221 A118 C491 -2.1349643 0.6623576 0.8377438 0.1656210
835 0.9996 0.1172 -0.1138 A605 C397 1.0336915 0.2140921 0.0921008 0.1122948
836 -2.3292 1.0997 0.8176 A71 C491 -2.1349643 0.6623576 0.8377438 0.2172406
837 0.0351 -1.1239 0.4830 A462 C556 0.6025305 -0.9291853 0.3583787 0.2955888
838 0.2351 2.8032 0.5822 A440 C100 -0.0270481 2.4238631 0.7389638 0.2660830
839 2.3001 0.9930 0.8629 A804 C160 2.3972612 0.8131370 0.6512993 0.1628750
840 -1.5402 -1.8112 0.9260 A194 C873 -1.6544946 -2.1236132 0.5230898 0.2765393
841 -1.5858 -2.0236 0.8210 A147 C873 -1.6544946 -2.1236132 0.5230898 0.1555393
842 0.0052 1.5450 -0.8905 A442 C342 -0.8373696 1.1920791 -0.7640059 0.4406615
843 0.3211 1.6671 0.9532 A513 C228 0.8994003 1.2236527 0.8271905 0.3825857
844 2.1852 0.5381 0.9681 A793 C160 2.3972612 0.8131370 0.6512993 0.2679663
845 2.8491 -0.8149 -0.2684 A888 C368 2.7225869 -0.3304495 -0.3889044 0.2438227
846 -0.3646 1.8874 -0.9970 A382 C342 -0.8373696 1.1920791 -0.7640059 0.4670282
847 0.5713 2.3279 0.9178 A547 C100 -0.0270481 2.4238631 0.7389638 0.2910491
848 0.4314 -1.2739 0.7556 A553 C556 0.6025305 -0.9291853 0.3583787 0.3043555
849 -0.4816 0.9482 0.3506 A378 C350 -0.5144181 1.0433207 0.4973778 0.0915722
850 -0.2094 -1.2389 -0.6687 A452 C655 -0.2597926 -1.1730025 -0.5212327 0.0879191
851 -0.3845 1.1105 0.5653 A392 C350 -0.5144181 1.0433207 0.4973778 0.0883399
852 -0.4227 -2.8208 0.5231 A429 C881 -0.5693053 -2.7024202 -0.2819962 0.3566938
853 2.2319 -1.8207 -0.4744 A873 C676 2.1950559 -1.7495404 0.0972307 0.2265448
854 0.7735 -0.8837 0.5642 A584 C556 0.6025305 -0.9291853 0.3583787 0.1407587
855 -0.8531 -1.4457 0.9470 A280 C710 -0.4532110 -1.5200102 0.8113151 0.2032947
856 -2.1654 0.1173 -0.9857 A139 C559 -1.4876296 -0.0265439 -0.7952792 0.3373450
857 -0.4133 0.9233 0.1519 A398 C350 -0.5144181 1.0433207 0.4973778 0.1888722
858 0.3064 -0.9570 0.0988 A525 C556 0.6025305 -0.9291853 0.3583787 0.1945080
859 0.4397 -1.3233 -0.7958 A535 C655 -0.2597926 -1.1730025 -0.5212327 0.3747858
860 0.6082 -2.9195 0.1879 A731 C763 0.7912866 -2.2489217 0.7766609 0.4808086
861 -1.9010 -0.3327 0.9975 A150 C728 -1.9816416 -0.8223302 0.8717976 0.2319914
862 -1.1718 2.2755 -0.8288 A219 C123 -0.9422671 2.4806302 -0.5791599 0.2281011
863 2.7476 0.7469 0.5312 A851 C160 2.3972612 0.8131370 0.6512993 0.1788917
864 -2.5761 -0.9133 -0.6800 A70 C766 -2.7284759 -0.4976307 -0.3415241 0.3021737
865 0.0124 -2.9452 -0.3265 A516 C881 -0.5693053 -2.7024202 -0.2819962 0.2896630
866 0.9754 -0.5410 0.4664 A628 C556 0.6025305 -0.9291853 0.3583787 0.2896920
867 -0.9695 -0.5161 -0.4323 A335 C536 -1.0323493 -0.1417128 0.3010375 0.3901913
868 -0.1966 -1.3076 -0.7354 A455 C655 -0.2597926 -1.1730025 -0.5212327 0.1373191
869 1.4617 -0.5118 0.8924 A697 C459 1.7910516 -0.5739146 0.9028013 0.1339558
870 -1.1555 -0.1048 0.5429 A271 C536 -1.0323493 -0.1417128 0.3010375 0.1339754
871 1.5318 -1.3975 -0.9973 A738 C508 1.3298237 -0.7586967 -0.8086783 0.3431338
872 -1.9226 -1.9712 0.6573 A91 C873 -1.6544946 -2.1236132 0.5230898 0.1849096
873 -1.9403 1.8255 0.7477 A117 C224 -1.4293520 1.9746821 0.7526839 0.2217046
874 -0.6796 1.2470 0.8147 A336 C350 -0.5144181 1.0433207 0.4973778 0.2287278
875 -0.1613 1.0129 -0.2252 A425 C287 0.5468719 1.0211322 -0.4670886 0.3194309
876 -0.6133 1.5167 -0.9314 A357 C342 -0.8373696 1.1920791 -0.7640059 0.2386949
877 -2.6175 -1.3702 -0.2983 A23 C766 -2.7284759 -0.4976307 -0.3415241 0.3422564
878 0.4819 -2.0337 0.9959 A609 C763 0.7912866 -2.2489217 0.7766609 0.2479492
879 -1.1432 -0.4086 -0.6183 A267 C559 -1.4876296 -0.0265439 -0.7952792 0.3011550
880 -1.0775 -1.2856 0.9465 A298 C710 -0.4532110 -1.5200102 0.8113151 0.3312947
881 -0.2063 1.2247 -0.6522 A432 C342 -0.8373696 1.1920791 -0.7640059 0.2584988
882 -0.3126 1.9148 0.9982 A405 C100 -0.0270481 2.4238631 0.7389638 0.3512837
883 1.9226 -2.2090 -0.3714 A864 C676 2.1950559 -1.7495404 0.0972307 0.4001821
884 0.2436 -1.6665 -0.9488 A555 C776 0.8378659 -2.2297098 -0.7821201 0.4413852
885 -2.5837 1.4946 0.1736 A6 C391 -2.3605439 1.3502882 -0.4417122 0.3275934
886 0.5240 -2.3070 -0.9307 A688 C776 0.8378659 -2.2297098 -0.7821201 0.1799120
887 -0.6504 0.7928 -0.2241 A371 C350 -0.5144181 1.0433207 0.4973778 0.3693268
888 0.7531 -1.0172 -0.6788 A610 C508 1.3298237 -0.7586967 -0.8086783 0.3217018
889 -0.7239 2.6588 -0.6550 A287 C123 -0.9422671 2.4806302 -0.5791599 0.1574590
890 0.0674 2.3913 -0.9199 A450 C81 0.5783356 2.3154644 -0.7661384 0.2468443
891 -1.5652 0.3799 -0.9211 A215 C559 -1.4876296 -0.0265439 -0.7952792 0.2032783
892 0.4389 -1.3194 -0.7928 A535 C655 -0.2597926 -1.1730025 -0.5212327 0.3722191
893 0.3523 -2.9723 0.1176 A666 C881 -0.5693053 -2.7024202 -0.2819962 0.5303604
894 -2.7257 -0.9674 0.4515 A30 C766 -2.7284759 -0.4976307 -0.3415241 0.4218564
895 -0.7362 2.8026 -0.4406 A250 C123 -0.9422671 2.4806302 -0.5791599 0.2221989
896 0.4608 0.9515 -0.3334 A509 C287 0.5468719 1.0211322 -0.4670886 0.0964642
897 -0.9475 -0.3572 -0.1585 A330 C536 -1.0323493 -0.1417128 0.3010375 0.2532913
898 0.8691 0.5017 -0.0834 A578 C397 1.0336915 0.2140921 0.0921008 0.2092334
899 -1.3751 0.1097 0.7841 A227 C536 -1.0323493 -0.1417128 0.3010375 0.3590754
900 -1.8520 -0.8448 -0.9994 A151 C802 -1.5935443 -1.4685237 -0.8727753 0.3362680
901 -1.6301 0.3771 -0.9451 A182 C559 -1.4876296 -0.0265439 -0.7952792 0.2319783
902 -1.0574 -1.1416 -0.8960 A318 C802 -1.5935443 -1.4685237 -0.8727753 0.2954309
903 -1.0258 -0.5064 -0.5169 A305 C536 -1.0323493 -0.1417128 0.3010375 0.3963913
904 0.7855 2.2340 0.9298 A627 C100 -0.0270481 2.4238631 0.7389638 0.3977491
905 -0.1088 -1.6831 -0.9496 A475 C655 -0.2597926 -1.1730025 -0.5212327 0.3631525
906 -0.4492 2.2182 0.9647 A363 C100 -0.0270481 2.4238631 0.7389638 0.2845170
907 0.4590 1.7579 -0.9831 A530 C81 0.5783356 2.3154644 -0.7661384 0.2979539
908 -0.4843 0.8750 -0.0129 A389 C350 -0.5144181 1.0433207 0.4973778 0.2362389
909 -1.2724 0.5443 0.7877 A258 C491 -2.1349643 0.6623576 0.8377438 0.3435552
910 -0.8224 -2.7464 -0.4986 A245 C881 -0.5693053 -2.7024202 -0.2819962 0.1712261
911 2.7113 -0.7916 0.5659 A886 C368 2.7225869 -0.3304495 -0.3889044 0.4757473
912 1.3150 -1.5734 0.9987 A730 C763 0.7912866 -2.2489217 0.7766609 0.4737581
913 -0.0896 -2.1376 0.9902 A507 C710 -0.4532110 -1.5200102 0.8113151 0.3866953
914 0.2347 1.5590 0.9059 A495 C228 0.8994003 1.2236527 0.8271905 0.3595857
915 -2.3118 -1.2366 -0.7832 A89 C802 -1.5935443 -1.4685237 -0.8727753 0.3465849
916 2.4548 -0.8983 -0.7894 A846 C368 2.7225869 -0.3304495 -0.3889044 0.4120444
917 -1.5621 -0.8875 -0.9791 A187 C802 -1.5935443 -1.4685237 -0.8727753 0.2395976
918 0.7140 -0.7037 -0.0701 A573 C556 0.6025305 -0.9291853 0.3583787 0.2551445
919 0.3019 -1.7019 0.9624 A527 C710 -0.4532110 -1.5200102 0.8113151 0.3626953
920 -0.4251 2.3952 0.9016 A281 C100 -0.0270481 2.4238631 0.7389638 0.1964504
921 2.3704 0.3195 -0.9200 A805 C175 1.9954496 0.9735682 -0.8296280 0.3731302
922 -2.4501 -0.5077 0.8648 A56 C728 -1.9816416 -0.8223302 0.8717976 0.2633621
923 -2.6727 1.3550 -0.0830 A9 C391 -2.3605439 1.3502882 -0.4417122 0.2251934
924 -1.9432 -0.4913 1.0000 A156 C728 -1.9816416 -0.8223302 0.8717976 0.1658914
925 -1.4864 1.9480 0.8929 A152 C224 -1.4293520 1.9746821 0.7526839 0.0746487
926 -0.1271 -2.7524 0.6553 A486 C881 -0.5693053 -2.7024202 -0.2819962 0.4764938
927 0.6923 -2.6805 -0.6399 A733 C776 0.8378659 -2.2297098 -0.7821201 0.2461920
928 -0.0360 2.5596 0.8286 A434 C100 -0.0270481 2.4238631 0.7389638 0.0781083
929 2.1069 -1.5891 0.7692 A824 C676 2.1950559 -1.7495404 0.0972307 0.3068552
930 2.5221 -1.1986 -0.6100 A854 C368 2.7225869 -0.3304495 -0.3889044 0.4299110
931 -0.9856 0.3668 0.3172 A304 C536 -1.0323493 -0.1417128 0.3010375 0.1904749
932 0.7638 2.8281 -0.3690 A569 C81 0.5783356 2.3154644 -0.7661384 0.3650795
933 -0.7350 -0.8900 0.5336 A374 C710 -0.4532110 -1.5200102 0.8113151 0.3965047
934 2.9613 0.3816 0.1681 A875 C160 2.3972612 0.8131370 0.6512993 0.4929250
935 1.1394 -2.7751 0.0131 A823 B4 1.3630182 -2.6647182 -0.0442455 0.1304485
936 2.9177 0.0183 -0.3970 A876 C368 2.7225869 -0.3304495 -0.3889044 0.1839861
937 1.2673 0.1687 0.6924 A659 C397 1.0336915 0.2140921 0.0921008 0.2930999
938 0.9708 2.5691 0.6655 A700 C100 -0.0270481 2.4238631 0.7389638 0.4055163
939 0.4998 0.8990 -0.2373 A519 C287 0.5468719 1.0211322 -0.4670886 0.1329976
940 -0.5049 0.8725 -0.1269 A389 C350 -0.5144181 1.0433207 0.4973778 0.2682055
941 0.5969 0.9729 -0.5127 A531 C287 0.5468719 1.0211322 -0.4670886 0.0479572
942 -2.6983 -1.2378 0.2484 A30 C766 -2.7284759 -0.4976307 -0.3415241 0.4534231
943 0.8018 1.1506 -0.8018 A585 C287 0.5468719 1.0211322 -0.4670886 0.2397024
944 2.3511 0.6178 -0.9024 A817 C175 1.9954496 0.9735682 -0.8296280 0.2613969
945 -0.8167 0.6568 -0.3061 A345 C342 -0.8373696 1.1920791 -0.7640059 0.3379515
946 -2.4149 -1.7734 0.0877 A52 B7 -2.3318600 -1.8725200 0.1143200 0.0695933
947 1.7972 -1.3389 -0.9705 A784 C508 1.3298237 -0.7586967 -0.8086783 0.4031338
948 -0.6392 -2.8255 -0.4422 A384 C881 -0.5693053 -2.7024202 -0.2819962 0.1177261
949 1.3753 -0.4742 -0.8383 A692 C508 1.3298237 -0.7586967 -0.8086783 0.1198649
950 -0.9870 0.4045 0.3591 A304 C536 -1.0323493 -0.1417128 0.3010375 0.2165415
951 2.4998 -0.3782 0.8491 A830 C459 1.7910516 -0.5739146 0.9028013 0.3193881
952 1.4232 -2.0919 -0.8479 A807 C776 0.8378659 -2.2297098 -0.7821201 0.2629746
953 0.2677 1.0791 0.4596 A492 C350 -0.5144181 1.0433207 0.4973778 0.2852251
954 1.2752 -1.1133 -0.9516 A703 C508 1.3298237 -0.7586967 -0.8086783 0.1840496
955 -0.9509 -1.1166 -0.8459 A342 C802 -1.5935443 -1.4685237 -0.8727753 0.3404811
956 -1.0314 -0.1408 -0.2833 A309 C536 -1.0323493 -0.1417128 0.3010375 0.1953999
957 -0.9190 1.8937 0.9945 A254 C224 -1.4293520 1.9746821 0.7526839 0.2777167
958 1.7216 0.3399 -0.9695 A732 C175 1.9954496 0.9735682 -0.8296280 0.3491299
959 0.7969 0.7094 -0.3597 A557 C287 0.5468719 1.0211322 -0.4670886 0.2230496
960 -0.9273 2.7901 0.3408 A192 C123 -0.9422671 2.4806302 -0.5791599 0.4147989
961 2.0903 0.3563 0.9927 A776 C160 2.3972612 0.8131370 0.6512993 0.3683997
962 2.0831 2.1314 0.1974 A836 C29 1.5896815 2.2890353 0.3101313 0.2545950
963 -0.2232 2.4002 0.9119 A363 C100 -0.0270481 2.4238631 0.7389638 0.1309170
964 1.5952 -2.3964 -0.4772 A847 C776 0.8378659 -2.2297098 -0.7821201 0.4096481
965 -1.0195 1.7248 1.0000 A233 C224 -1.4293520 1.9746821 0.7526839 0.3023501
966 2.6734 0.1140 -0.7371 A838 C368 2.7225869 -0.3304495 -0.3889044 0.2806107
967 1.7650 -1.6558 0.9075 A796 C459 1.7910516 -0.5739146 0.9028013 0.3708786
968 0.8482 -0.6181 -0.3108 A599 C508 1.3298237 -0.7586967 -0.8086783 0.3733662
969 0.0026 -1.0002 -0.0210 A474 C655 -0.2597926 -1.1730025 -0.5212327 0.3118092
970 1.0801 -0.2942 0.4739 A639 C397 1.0336915 0.2140921 0.0921008 0.3121666
971 -0.0106 2.5606 -0.8281 A400 C81 0.5783356 2.3154644 -0.7661384 0.2986776
972 0.8477 1.6050 0.9827 A572 C228 0.8994003 1.2236527 0.8271905 0.1961857
973 0.3534 -1.6079 0.9354 A539 C710 -0.4532110 -1.5200102 0.8113151 0.3395286
974 0.4017 1.1145 -0.5790 A501 C287 0.5468719 1.0211322 -0.4670886 0.1168170
975 1.3725 -0.7583 -0.9019 A713 C508 1.3298237 -0.7586967 -0.8086783 0.0454315
976 -0.0271 0.9997 0.0067 A449 C350 -0.5144181 1.0433207 0.4973778 0.3405389
977 -0.9028 -0.4593 0.1600 A350 C536 -1.0323493 -0.1417128 0.3010375 0.1960580
978 -0.5592 2.9106 0.2664 A251 C100 -0.0270481 2.4238631 0.7389638 0.4971509
979 -0.9973 -2.4039 -0.7981 A232 C881 -0.5693053 -2.7024202 -0.2819962 0.4142062
980 0.3857 2.0988 0.9910 A547 C100 -0.0270481 2.4238631 0.7389638 0.3299491
981 -2.3473 0.7687 -0.8827 A92 C391 -2.3605439 1.3502882 -0.4417122 0.3452733
982 0.5050 1.1216 -0.6381 A528 C287 0.5468719 1.0211322 -0.4670886 0.1044504
983 -0.7064 -0.8184 -0.3946 A376 C655 -0.2597926 -1.1730025 -0.5212327 0.3092809
984 2.0318 -0.7054 0.9886 A779 C459 1.7910516 -0.5739146 0.9028013 0.1526775
985 -1.1852 1.2823 -0.9672 A237 C342 -0.8373696 1.1920791 -0.7640059 0.2137485
986 -1.4971 0.2476 -0.8759 A228 C559 -1.4876296 -0.0265439 -0.7952792 0.1214117
987 -0.8561 -0.6772 -0.4180 A335 C655 -0.2597926 -1.1730025 -0.5212327 0.3984475
988 -0.8928 -0.4866 0.1827 A350 C536 -1.0323493 -0.1417128 0.3010375 0.2009246
989 -0.3445 2.0960 -0.9923 A365 C81 0.5783356 2.3154644 -0.7661384 0.4561539
990 -2.9348 -0.2855 -0.3164 A5 C766 -2.7284759 -0.4976307 -0.3415241 0.1478596
991 -2.7377 -0.2260 0.6648 A45 C766 -2.7284759 -0.4976307 -0.3415241 0.4290596
992 0.7107 2.5872 -0.7304 A618 C81 0.5783356 2.3154644 -0.7661384 0.1466128
993 -1.0415 -0.2291 0.3582 A290 C536 -1.0323493 -0.1417128 0.3010375 0.0512335
994 -1.0222 -2.4375 0.7657 A210 C873 -1.6544946 -2.1236132 0.5230898 0.3962639
995 2.8401 0.8724 0.2388 A852 C160 2.3972612 0.8131370 0.6512993 0.3048670
996 0.5626 0.8335 -0.1057 A532 C287 0.5468719 1.0211322 -0.4670886 0.1882496
997 -1.5951 0.0871 -0.9154 A212 C559 -1.4876296 -0.0265439 -0.7952792 0.1137450
998 0.8443 -2.0716 -0.9715 A676 C776 0.8378659 -2.2297098 -0.7821201 0.1179746
999 2.2937 0.8478 0.8954 A804 C160 2.3972612 0.8131370 0.6512993 0.1274416
1000 1.4904 -0.4691 0.8992 A697 C459 1.7910516 -0.5739146 0.9028013 0.1363558

hist(act_pred$diff, breaks = 30, col = "blue", main = "Mean Absolute Difference", xlab = "Difference")
Figure 22: Mean Absolute Difference

Figure 22: Mean Absolute Difference

7. Executive Summary

8. References

  1. Topology Preserving Maps : https://users.ics.aalto.fi/jhollmen/dippa/node9.html

  2. Vector Quantization : https://en.wikipedia.org/wiki/Vector_quantization

  3. K-means : https://en.wikipedia.org/wiki/K-means_clustering

  4. Sammon’s Projection : https://en.wikipedia.org/wiki/Sammon_mapping

  5. Voronoi Tessellations : https://en.wikipedia.org/wiki/Centroidal_Voronoi_tessellation